SQC25

7/21/2019 SQC25

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Session Numbers

LCM & HCF – I

Numbers is one of the most important topics for CAT and other management entrance exams, questionsfrom which have appeared consistently and in significant numbers in all these exams.

Key concepts discussed:

HCF of two or more numbers is the greatest number which completely divides the numbers. In other

words, it is the greatest factor common to the numbers.

• The largest number N, which divides the numbers n1, n

2 and n

3 (n

1 > n

2 > n

3) and leaves a remainder

of ‘r’ in each case is given by { }1 2 2 3N = HCF (n n ), (n n )− − .

• The largest number N, which divides the numbers n1, n

2 and n

3and leaves remainders of r

1,r2and r

3

respectively is given by { }1 1 2 2 3 3N = HCF (n r ),(n r ),(n r )− − − .

If there are three types of items such that number of items of first type is n1, that of second type is

n2 and that of third type is n

3, then the minimum number of groups (N) in which these items can be

slotted such that all the groups have same number of items and each of the groups has only one

type of item 31 2

1 2 3 1 2 3 1 2 3

nn nN

HCF (n , n , n ) HCF (n , n , n ) HCF (n , n , n )= + + 1 2 3

1 2 3

n n n

HCF (n , n , n )

+ += .

• The LCM of two or more numbers is the least number which is divisible by all the numbers.

• If p q rN a b c ....= × × × is a natural number such that N = LCM (x, y), then the number of distinct

pairs (P) of x and y is given by(2p 1)(2q 1)(2r 1)... 1

P2

+ + + += .

• LCM and HCF of fractions

a c e LCM (a, c, e)LCM , ,

b d f HCF (b, d, f )

=

;

a c e HCF (a, c, e)HCF , ,

b d f LCM (b, d, f )

=

, where

a c e, and

b d fare the fractions in simplest form.

LCM (N1, N

2) × HCF (N

1, N

2) = N

1 × N

2

The above relation is true for more than two numbers if and only if each of numbers is co-prime to

rest of the numbers i.e. HCF of any two numbers is 1. Note that, if the HCF of a set of numbers is

equal to 1, the above relation is not necessarily valid.

• 1 2 1 2HCF N + N , LCM (N ,N ) = HCF (N1, N

2)

7/21/2019 SQC25


SessionNumbers

• The number N, which when divided by n1, n

2 and n

3 leaves the same remainder of ‘r’ in each case, is

given by 1 2 3N k LCM(n , n , n ) r= × + , where k is a whole number.

• The number N, which when divided by n1, n

2 and n

3 leaves remainders of r

1,r2and r

3 respectively such

that n1 – r

1 = n

2 – r

2 = n

3– r

3= d, is given by 1 2 3

N k LCM(n , n , n ) d,= × − where k is a natural number.

• The number N, which when divided n1 and n

2 leaves remainders of r

1and

r2

such that 1 2r r≠ and

1 1 2 2n – r n – r ,≠ is given by 1 2 LN k LCM(n , n ) N ,= × + where k is a whole number and NL is the

smallest number which satisfies the aforementioned conditions.

Three bells B1, B2 and B3 chime at regular intervals of T1, T2 and T3 min respectively. These bells

chime for c1, c

2 and c

3 min respectively. If all the bells start chiming together, then the time interval

(I) between two successive occasions when they start chiming together again is given by

I = LCM (T1 + c1, T2 + c

2, T3 + c

3) min. If the bells chime for negligible amounts of time, then

I = LCM (T1, T2, T3) min.

Highlight: This session deals with application based questions of LCM & HCF which are of moderatedifficulty level.

7/21/2019 SQC25


Session Numbers

The questions discussed in the session are given below along with their source.

Q1. a, b, c d and e are integers such that 1 a b c d e.≤ < < < < If a, b, c, d and e are in geometricprogression and lcm (m , n ) is the least common multiple of m and n , then the maximum value of

1 1 1 1is

lcm(a,b) lcm(b,c) lcm(c,d) lcm(d,e)+ + +

(a) 1 (b)15

16(c)

79

81(d)

7

8(d) None of the above

(XAT 2010)

Q2. Which is the least number that must be subtracted from 1856, so that the remainder when dividedby 7, 12 and 16 is 4.(a) 137 (b) 1361 (c) 140 (d) 172

(CAT 1994)

Q3. A number which when divided by 10 leaves a remainder of 9, when divided by 9 leaves a remainderof 8, by 8 leaves a remainder of 7, etc., down to where, when divided by 2, it leaves a remainder of1, is :(a) 59 (b) 419 (c) 1259 (d) 2519

(FMS 2011)

Q4. Three bells chime at an interval of 18 min, 24 min and 32 min. At a certain time they begin to chime

together. What length of time will elapse before they chime together again?

(a) 2 hr and 24 min (b) 4 hr and 48 min

(c) 1 hr and 36 min (d) 5 hr

(CAT 1995)

Q5. Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. The number ofboxes containing the same number of oranges is at least(a) 5 (b) 103 (c) 6 (d) Cannot be determined

(CAT 2001)

Q6. For two positive integers a and b define the function h(a, b) as the greatest common factor (G.C.F)

of a, b. Let A be a set of n positive integers. G(A), the G.C.F of the elements of set A is computed by

repeatedly using the function h. The minimum number of times h is required to be used to compute

G is

(a)2

1 n (b) (n – 1) (c) n (d) None of these

(CAT 1999)

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