ORIGINAL ARTICLE

Stijn De Rammelaere á Els StuyvenAndre Vandierendonck

The contribution of working memory resourcesin the veri®cation of simple mental arithmetic sums

Received: 16 April 1998 /Accepted: 20 July 1998

Abstract The present study replicated the investigationsof Lemaire, Abdi, and Fayol with some modi®cations:the random time interval generation (RIG) task wasused and the stimuli were created in another way. Theresults provide additional evidence for the crucial role ofthe central executive in the speed of solving both trueand false sums and for the role of the phonological loopin solving false sums. However, the ®ndings concerningthe role of this slave system in solving true sums weredi�erent. Possible explanations and limitations of theseresults are discussed.

Introduction

Mental arithmetic is an important everyday skill ofmany adults. Solving simple arithmetic problems (suchas 7� 6 � ?) is also a key component of elementaryeducation. Research so far has revealed two major de-terminants of performance: the organization of simplearithmetic facts in long-term memory and the processingof the information in working memory. The ®rst topichas attracted many researchers and has been docu-mented fairly extensively (e.g., Anderson, 1983). Therole of working memory in arithmetic, however, hasreceived much less attention, and studies have mainlyfocused on di�cult problems, such as 435� 287 or13� 18� 13� 21� 13. The objective of the presentstudy was to further explore the role of working memoryin simple arithmetic.

Most researchers agree that arithmetic facts arestored in an interrelated network in long-term memory

(e.g., Anderson, 1983; Ashcraft & Fierman, 1982;Campbell & Graham, 1985; Lemaire & Siegler, 1995).The general notion is that adults and older childrenpossess a vocabulary of known simple arithmetic oper-ations (such as addition, multiplication, and division)which are organized in the form of an associative se-mantic network. For example, the answer to the prob-lem 8� 2 is usually known without having to follow anyform of calculation algorithm. The fact that these as-sociative models explain fairly well four typical e�ects ofmental arithmetic resulted in a reasonable consensusabout the validity of these kinds of models.

A ®rst e�ect is the problem-size/di�culty e�ect.Simple arithmetic problems become more di�cult as thesize of the operands increases (e.g., 2� 3 in comparisonto 14� 28; see Ashcraft & Battaglia, 1978; Groen &Parkman, 1972). Secondly, there is the split e�ect.Arithmetic problems with false answers that are close tothe correct answer are more di�cult than problems withfalse answers that are more discrepant (e.g., 2� 3 � 6 incomparison with 2� 3 � 10; e.g., Ashcraft & Battaglia,1978; Zbrodo� & Logan, 1990). The associative-confu-sion/interference e�ect is a third observation. When apresented false answer matches a correct answer underanother arithmetic operation, error rates are higher andlatencies are longer (e.g., 7� 5 � 35 and 7� 5 � 12;e.g., Le Fevre & Kulak, 1994; Lemaire, Barrett, Fayol, &Abdi, 1994). Finally, there is the odd-even e�ect. Par-ticipants take longer to reject a false answer that is even(odd) when the correct answer is even (odd) (e.g.,Krueger, 1986).

In contrast to these long-term memory studies,working memory research with arithmetic problems hasfocused heavily on complex problems. For example,Hitch (1978) found that participants divide a problemsuch as 547� 86 into elementary stages, and that themost important sources of errors are due to (a) forget-ting partial results on calculation (such as 7� 6 in ourexample) and (b) forgetting initial information.

In the exploration of the role of working memoryin mental arithmetic, the working memory model of

Psychological Research (1999) 62: 72±77 Ó Springer-Verlag 1999

E. Stuyven (&) á S. De Rammelaere á A. VandierendonckDepartment of Experimental Psychology,University of Ghent,Henri Dunantlaan 2, B-9000 Ghent, Belgium;Tel.: +32-(0)-9-264 64 36; Fax: +32-(0)-9-264 64 96;e-mail: Els.Stuyven@rug.ac.be

Baddeley and Hitch (1974) has already proven to be auseful framework (e.g., Lemaire, Abdi, & Fayol, 1996;Logie & Baddeley, 1987; Logie, Gilhooly, & Wynn,1994). According to Baddeley and Hitch, workingmemory refers to the temporary storage and processingof information in a variety of cognitive tasks (Baddeley,1986, 1990; Baddeley & Hitch, 1974). Their model givesa multi-componential description of working memory,with three components: the central executive, the pho-nological loop, and the visuo-spatial sketch pad. Thecentral executive is a limited-capacity system that,among others, monitors the allocation of mental re-sources to the two slave systems during cognitive tasks.

Logie et al. (1994) used this framework to investigatethe role of the di�erent systems of working memory incomplex arithmetic problems. They used a calculationtask in which participants had to add a series of two-digit numbers that were auditorily or visually presentedwhile the di�erent components of working memory wereloaded. Logie et al. found that (a) the role of the pho-nological loop is probably to keep track of running to-tals and to maintain accuracy in calculation, and that (b)the role of the central executive is most likely to performthe calculations required for mental addition and toproduce approximately correct answers.

Lemaire et al. (1996) investigated the role of workingmemory in single, simple arithmetic problems. Theseresearchers used the veri®cation task (e.g., 8� 4 � 12True? False?) and investigated only one-digit numbers.Although Lemaire et al. also investigated multiplication,only their results of the single sums concern us here. Thefalse answers were created in two ways. First, the falseanswer in a ``confusion-problem'' was the product of thetwo terms (e.g., 5� 3 � 15). The false answer in a ``non-confusion problem,'' on the contrary, was the productplus or minus 1 (e.g., 5� 3 � 14 or 16), in order toequate the splits. In addition to a control condition therewas also a condition with articulatory suppression and acondition with random letter generation. Lemaire et al.found greater disruption with correct equations (e.g.,3� 5 � 8) when (a) the phonological loop was loaded(by means of articulatory suppression) and when (b) thecentral executive was loaded (by means of random lettergeneration). However, greater disruption with falseequations (e.g., 3� 5 � 15 or 14 or 16) was only foundwhen concurrent random letter generation was per-formed. In other words, this suggests that the phono-logical loop is only involved in solving correct equationsbut not in solving false equations. The authors con-cluded that the central executive is surely a critical sys-tem involved in simple mental arithmetic, for both trueand false equations.

Despite the very interesting ®ndings of Lemaire et al.(1996), some important problems have remained un-solved.

(1) Is it really the case that the phonological loop isonly involved in true and not in false sums? This ispossible in view of the observation that basic arithmeticfacts ± about true equations; but not about false ones ±

are ®rst learned by means of oral repetition. However,there are some methodological remarks that limit this®nding. First, Lemaire et al. (1996) used only theproduct of the two terms (confusion problem) or theproduct plus or minus 1 (non-confusion problem) inorder to get false answers. In such a way, however, theyinvestigated mainly ``extreme'' false answers (e.g.,9� 8 � 72 or 71 or 73; 6� 7 � 42 or 41 or 43) that,because of the split e�ect, are relatively easy to solve.False answers closer to the correct ones are needed to beable to generalize the observation that the phonologicalloop is not involved in solving false equations. Second,the participants of Lemaire et al. were asked to say``the'' every two seconds. If it is the case that partici-pants take this task very seriously and monitor the timeelapsed since the last articulation, this task can be seennot as an automatic task, but as a task requiring centralexecutive resources in order to be performed accurately.

(2) The random letter generation task not only loadsthe central executive, but also interferes with the pho-nological loop. As a result, clean-cut conclusions andhard evidence are not easy to obtain. Vandierendonck,De Vooght, and Van der Goten (1998) have developedand tested a central executive task of which the load onthe slave systems is too small to have detectable e�ects.In this random time interval generation (RIG) task,participants are to tap a random (i.e., an unpredictable)rhythm. The requirement to be random and to avoidautomaticity loads the central executive, whereas there isneither empirical nor logical ground to assume that thereis interference with a slave systems. This task has alreadybeen shown to a�ect span, supraspan, backward span(Vandierendonck et al., 1998), stimulus-independentthoughts (Stuyven & Van der Goten, 1995), latencies ofsaccadic eye movements (Stuyven, Van der Goten,Vandierendonck, Claeys, & Crevits 1999), and a dualcontent location visual span task (Martein, Kemps,& Vandierendonck, in press).

Method

The aim of this experiment was to explore whether theconclusions of Lemaire et al. (1996) still hold, if someimportant modi®cations, which will become clear later,are introduced.

Participants. Forty ®rst-year psychology students (13 males, 27females) of the University of Ghent in Belgium participated forcourse requirements and credit, volunteering for this particularexperiment. The mean age of the participants was 18.8 (range:17.8±20.7) years. All the participants had normal or corrected-to-normal vision.

Stimuli. The stimuli were single sums presented in standard form(i.e., a + b � c). The terms a and b were always one-digit numbers.Other stimuli than those of Lemaire et al.'s (1996) were chosen: (1)true sums, such as 8� 4 � 12; (2) split � +1, such as 8� 4 � 13,and (3) split � +5, such as 8� 4 � 17. These sizes of the splitsenabled us to investigate whether the conclusions of Lemaire et al.

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(1996) still hold when the sums are combined with the smallest splitpossible (+1) and when they are combined with a bigger but not``extreme'' split (+5). All the sums were controlled for the numberof carries: the c term was, for all three kind of sums, a number from10 to 19 (carry � 1). Because of this, no negative splits were in-cluded. If we had worked with both a positive split (e.g.,8� 4 � 17) and a negative split (e.g., 8� 4 � 7), the carries wouldhave no longer been controlled. Just like in the study of Lemaire etal., the numbers 0 and 1 were omitted because there is evidence thatthese problems (e.g., x + 0 or x + 1) are not solved by retrievingthe solution directly from memory but instead by retrieving rules(e.g., x + 0 � x) that guide their solution (see Ashcraft, 1982;Baroody, 1985).

The sums were further controlled as follows: (1) half of thecorrect results of the sums were even, the other half were odd; (2)for half of the sums, the ®rst number was bigger than the second(a > b), for the other half the inverse was true (a < b), and (3) thec term was never the product of a ´ b, in other words, the asso-ciative-confusion/interference e�ect could not play a role in theveri®cation of the sums.

In this way, 24 combinations of the form a + b were formed. Inone condition, these were once presented with a split of +1, oncewith a split of +5, and twice with the correct solution. As a result,every series consisted of 96 (4 ´ 24) trials, namely 48 (2 ´ 24) trueand 48 (2 ´ 24) false single sums. Before the presentation of eachseries, the sequence of all sums was randomized. Only in less than0.5% of the trials was the sum presented the same as the one before.

Procedure and design. The stimuli were presented horizontally inthe center of a computer screen, in yellow with a black background.The equations remained on until the participant responded, unlessthere was still no response after 10 s. The participants were in-structed to solve the sums as accurately and as fast as possible bypressing the appropriate key. The left and the right button of themouse were designated as true and false. For half of the partici-pants, the left button was designated as true, and for the other half,it was designated as false. All participants were instructed to usetheir fore®nger and their middle ®nger of the right hand to pressthese keys. The inter-trial interval was 1 s.

Each participant participated under every condition, in contrastto Lemaire et al. (1996), where the participants participated underonly one condition. There were four conditions: control, articula-tory suppression, random letter generation, and random time in-terval generation.

In the control (CON) condition the participants solved the sumswithout a secondary task. The articulatory suppression (AS) con-dition required the participants to say ``the'' (``de'' in Dutch) aloudand quickly while they were solving the sums. This secondary taskwas meant to load the phonological loop, and only the phono-logical loop. In contrast of Lemaire et al. (1996), the participantsdid not say ``the'' every 2 s, but did so continuously and withoutstopping. This modi®cation was introduced because of the reasonmentioned before.

In the random letter generation (RLG) condition, the partici-pants were required to say one random letter of the alphabet at arate of one letter per second while they were solving the sums. Therate was indicated by a metronome that continued throughout theseries. The participants were instructed to avoid stereotypical se-quences (e.g., ``a-b-c-d'' or ``o-p-q-r'') or spelling out words (e.g.,``c-a-t''). Thus, the standard version of RLG was used in this study,in contrast to the investigation of Lemaire et al. (1996), where amodi®ed version was used, namely one random letter of the series``a-b-c-d-e-f'' at a rate of one per two seconds. Despite the alreadymentioned drawback, this secondary task was introduced because(a) the results of this condition should serve as a baseline for thefourth condition, (b) it is one of the most frequently used tasks toload the central executive, and (c) it would maintain the compati-bility with the study of Lemaire et al.

The random time interval generation task (Vandierendonck etal., 1998) was used as a fourth condition; the participants wereasked to tap an unpredictable rhythm on the zero key of thenumeric keypad while they were solving the sums. They were

instructed to use their left fore®nger and were told that the rhythmhad to be as random and unpredictable as possible. This task wasmeant to load the central executive and not to interfere with one ofthe slave systems, as in the previous secondary task. In order toobtain a measure of randomness, the tap sequences of the partici-pants were registered.

Four conditions imply 24 possible sequences of these conditions(4! � 24). The ®rst 24 participants each participated in one of thesesequences. The sequences for the next 16 participants were selectedat random, but with the restriction that no sequence was executedby more than two participants. As a result, each sequence wasexecuted by (approximately) the same number of participants, andpossible e�ects of learning or boredom were eliminated. The par-ticipants were permitted a 3-min rest period between the condi-tions.

At the start of the experiment, the participants solved 16 ran-dom practice trials (8 true sums, 4 split � 1, 4 split � 5) in order tofamiliarize themselves with the apparatus, the procedure, thestimulus display, and the response keys. After each practice trial,the participants received feedback: according to the answer given,the text ``correct answer'' or ``wrong answer'' was given at thebottom of the screen for 1 s. After these practice trials, no morefeedback was given, and the already mentioned sequence of con-ditions was followed. Each series started with a ®xation point (``!'')in the middle of the screen that remained for 500 ms. In theexperimental conditions, the participants were ®rst required topractise the concurrent task until they felt comfortable with itand showed no apparent problems. They were told that is wasimportant not to stop performing the secondary task while theywere solving the sums. If they stopped, which happened only once,this condition was rerun all over. The participants were testedindividually in a quiet room. Each experimental session lastedapproximately 25±35 min.

A 4 (Load: CON, AS, RLG, RIG) ´ 3 (Sum: true, split � +1,split � +5) within-participants design was used. This design waspreferred to a between-participants design in order to eliminatedi�erences in arithmetic skills between the participants. Thus, everyparticipant solved three kinds of sums under each condition. As aresult, all participants solved exactly 400 sums [16 practice tri-als + (4 ´ 96)] experimental trials.

Results

The data were analyzed by means of a multivariateanalysis with contrasts between the 12 dependent vari-ables: the mean latencies per participant under thecombinations of condition and kind of sum, or theproportion of correct responses per participant underthe combinations of condition and kind of sum. Thisanalysis conforms to the suggestions of McCall andAppelbaum (1973) for a correct analysis of repeated-measures designs.

Randomness analyses

Every sequence of random taps1 was analyzed by meansof the method described by Vandierendonck et al. (1998)and more in detail in Vandierendonck (1998) in order to

1A random analysis of the random letter generation task was notpossible because the most straight-forward statistic is the one de-scribed by Evans (1978). However, this statistics requires exactly262 � 676 observations. The trial block did not last long enough toproduce that many digits. However, there is no reason to assumethat the task was not performed accurately.

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®nd out whether participants complied with task in-structions. In essence, random time intervals can beconverted into a series of binary events. To that end, thecomplete time course is subdivided into a sequence of®xed intervals, each of which either contains an event (akeypress) or does not. By means of appropriate statis-tics, the degree of statistical independence and the ten-dency to alternate (or to perseverate, if such arose) canbe estimated. Six participants deviated from random-ness. The data analyses reported below were performed,with and without these six participants. As there were nodi�erences in the pattern of results, the analyses on thecomplete data are reported here. By means of theseanalyses, a median-split was also introduced: the par-ticipants were divided into a group with the 20 mostrandom sequences and a group with the 20 least randomsequences. The latencies and the proportions of correctanswers of these two groups were analyzed and com-pared. As there were no di�erences in the pattern ofresults between the two groups, this analysis will not bereported.

Latencies

The trials in which the participants made errors weredropped from the analysis. The mean latencies and theirstandard deviations are presented in Table 1. The maine�ect of condition was signi®cant, F(3,37) � 89.44,p < .001, as was the main e�ect of kind of sum,F(2,38) � 41.61, p < .001, but their interaction wasnot: F(6,34) � 1.64, p > .15. In contrast to the controlcondition, AS had no e�ect, F(1,39) � 1.38, p> .20, butRLG and RIG did have an e�ect: F(1,39) � 206.07,p < .001 and F(1,39) � 34.21, p < .001, respectively.Also, the e�ect of RIG was signi®cantly di�erent fromAS, F(1,39) � 41.65, p < .001, and from RLG,F(1,39) � 195.77, p < .001.

Across all conditions, there was a signi®cant splite�ect, i.e., the sum with split � 5 (weighted M=1628ms) were solved signi®cantly faster than sums with split=1 (weighted M=1833 ms): F(1,39) � 68.67, p < .001.The latencies of true sums (weightedM � 1588 ms) weresigni®cantly shorter than the latencies of sums withsplit � 1, F(1,39) � 66.95, p < .001, but did not di�er

signi®cantly from sums with a split of 5, F(1,39) � 1.69,p > .20.

Accuracy

The mean proportions of correct answers and theirstandard deviations are presented in Table 2. As can beseen, accuracy was very high. Consequently, these dataare not very sensitive, and di�erences in accuracy mustbe interpreted with caution.

The main e�ect of condition was signi®cant,F(3,37) � 15.29, p < .001, as was the main e�ect of thekind of sum, F(2,38) � 41.33, p < .001, but their in-teraction was not: F(6,34) � 1.48, p > .20. Across alltrials, AS had no e�ect, F(1,39) � 2.15, p > .15; on theother hand, RIG and RLG did have an e�ect:F(1,39) � 4.07, p � .05, and F(1,39) � 46.85, p < .001,respectively. Also, the e�ect of RIG was signi®cantlydi�erent from RLG, F(1,39) � 16.62, p < .001, but didnot di�er signi®cantly from AS, F(1,39) � 1.58, p> .20.

Across all conditions, there was a signi®cant splite�ect, i.e., the sums with split � 5 (M=0.96) weresolved signi®cantly better than sums with split � 1(M=0.86): F(1,39) � 81.81, p < .001. True sums(M=0.91) were solved signi®cantly worse than sumswith split � 5, F(1,39) � 29.04, p < .001, and signi®-cantly better than sums with a split of 1,F(1,39) � 24.75, p < .001.

Correlation between latencies and accuracy

In order to test for a possible trade-o� between latenciesand accuracy, correlations between the two dependentmeasures were calculated for each condition. None ofthem was signi®cant (CON: .15; AS: .17; RLG: .14; andRIG .01).

Discussion

The aim of the present study was to explore whether theconclusions of Lemaire et al. (1996) still hold if someimportant modi®cations are introduced. Other stimuli

Table 1 Mean latencies (in ms) per condition and per kind ofsum.a Standard deviations are given between parentheses.AS � articulatory suppression; RLG � random letter generation;RIG � random time interval generation

Control AS RLG RIG

True 1203 1260 2552a 1415a

(315) (363) (695) (396)Split � 1 1455 1481 2778a 1659a

(349) (422) (787) (412)Split � 5 1183 1239 2670a 1466a

(281) (356) (726) (426)

a signi®cant in comparison to control on p < :001

Table 2 Mean proportions correct responses per condition and perkind of sum.a Standard deviations are given between parentheses.AS � articulatory suppression; RLG � random letter generation;RIG � random time interval generation

Control AS RLG RIG

True 0.94 0.92a 0.86a 0.93(0.05) (0.06) (0.11) (0.06)

Split � 1 0.87 0.87 0.83a 0.86(0.08) (0.10) (0.11) (0.09)

Split � 5 0.98 0.97 0.93a 0.95a

(0.03) (0.03) (0.07) (0.08)

a signi®cant in comparison to control on p < .05

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(i.e., other false answers) were used, a new secondarytask, the random time interval generation task (Van-dierendonck et al., 1998), was applied, and standardimplementations for the other secondary tasks, namelyarticulatory suppression and random letter generation,were used.

The ®rst question concerned the role of the phono-logical loop in both true and false sums. We found thatarticulatory suppression had no e�ect and the interac-tion of condition by type of sum was not signi®cant,suggesting that the secondary tasks did not play a dif-ferent role for the three kinds of sums. This implies thatin this experiment the phonological loop was not in-volved in true nor false sums. For the latter sums, the®ndings of Lemaire et al. (1996) was replicated in thisstudy, but this time with other false answers. The pho-nological loop is apparently not involved in solving``extreme'' false sums (Lemaire et al., 1996), nor in falsesums with splits of 1 and 5. In other words, this studyprovides additional empirical evidence for the conclu-sion that the phonological loop is not involved in solvingfalse sums. However, the ®ndings concerning the truesums were di�erent; in this experiment we found that aload on the phonological loop had no e�ect on the la-tencies of true sums, in contrast to the ®ndings of Le-maire et al. (1996).

How can such apparently contradictory results beexplained? One possible explanation is the di�erence inthe selected stimuli: in this study, the sums were con-trolled for the number of carries. As a result, manypossible one-digit sums were not studied. It is not im-possible that the phonological loop plays a di�erent rolefor these sums. However, a second possible explanationsounds more plausible: the articulatory suppression, asused in the study of Lemaire et al. (1996), probably alsointerfered with the central executive, because the par-ticipants were instructed to say ``the'' at a rate of exactlyone every 2 seconds. There is some recent empericalevidence for this suggestion: Stuyven et al. (1999) founda di�erent e�ect of ®xed tapping on saccadic eyemovements under strict and under lenient instructions.In this experiment, ®xed tapping was a secondary taskwhich required participants to tap a button every secondand which, just like articulatory suppression, does notcall on resources of the central executive. However,when strict instructions were given (``Tap at a rate ofexactly one per second''), this secondary task was foundto require central executive resources. It has to be em-phasized that this possible explanation is only a sug-gestion. Future investigations will be needed to clarifythe role of the phonological loop in solving true sums.

The second question concerned the contribution ofthe random time interval generation task in the ®eld ofmental arithmetic. If we assume that the central execu-tive indeed plays a crucial role in the latencies of allkinds of sums, as Lemaire et al. (1996) found, then ananalogous e�ect of this secondary task which claims toload only the central executive is expected. This wasexactly what we found for the three kinds of sums. Thus,

this study provides additional empirical evidence for thecrucial role of the central executive in the latencies of allkinds of sums. These ®ndings not only replicate but alsoextend the results of Lemaire et al. (1996). First, thesee�ects were found by means of a secondary task thatloads only the central executive and does not interferewith one of the slave systems. Second, the crucial rolehere was also demonstrated with additional false an-swers, namely splits of 1 and 5. It is also important tomention that RIG had more subtle and sensitive e�ectsin comparison to random letter generation, where la-tencies were almost always doubled. The RIG taskadded between 200 and 250 ms, but nevertheless reachedconvincing signi®cance (for both latencies and accura-cy). In short, this study demonstrates the usefulness ofthe RIG task in this ®eld of research.

A last remarkable ®nding of this study is that truesums were not solved signi®cantly faster than sums witha split of 5 (in fact, accuracy of true sums was worsethan the accuracy of sums with a split of 5 ). This goesagainst the general notion that ``latencies for true sumsare typically shorter than latencies for any false prob-lems because true problems involve stronger operand-correct answer associations'' (Lemaire et al., 1996).Future studies should investigate the robustness of this®nding and, if replicated, models of arithmetic repre-sentation in long-term memory will have to take it intoaccount.

The most important limitations of this study aretwofold. First, not all possible single sums were studied.Consequently, it was not possible to partition the sumsinto ``easy'' and ``di�cult'' sums, because all the sumshad a carry of ``1.'' However, as the study of Lemaire etal. (1996) demonstrates, such a distinction can revealinteresting e�ects. Second, there were no single-taskcontrol conditions, so it was not possible to investigatepossible trade-o� e�ects between primary and secondarytask performance. Nevertheless, as pointed out in thisdiscussion, we believe that this study contributes a fewinteresting ®ndings to this ®eld of research.

Acknowledgements The research reported in this article was sup-ported by a Ph.D. Grant to Els Stuyven (BOF No. 011D0896) andthe Belgian program on ``Inter-university Poles of Attraction''Grant No. P4/9 (1997±2001) from the Department of SciencePolicy to Andre Vandierendonck.

We are indebted to Eva Kemps for useful remarks, Jacky Lillyfor checking and improving the English, and Antoine Tavernier fortechnical support.

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