7/21/2019 SQC20

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Session Numbers

Digit based

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:

ab = 10a + b, where ab is a two-digit number; abc = 100a + 10b + c, where abc is a three-digit

number.

• ab + ba = (10a + b) + (10b + a) = 11(a + b) i.e. the sum of a two-digit number and the number formed

by interchanging its digits is always divisible by 11.

• ab – ba = (10a + b) –(10b + a) = 9(a – b) i.e. the difference of a two-digit number and the number

formed by interchanging its digits is always divisible by 9.

• abc – cba = (100a + 10b + c) – (100c + 10b + a) = 99(a – c) i.e. the difference of a three-digit

number and the number formed by interchanging its digits is always divisible by 99.• abc – (a + b + c) = 99a + 9b i.e. the difference between a number and the sum of its digit is always

divisible by 9.

• The digit sum of a number is the sum of digits of the number. Example: digit sum of 365 is

(3 + 6 + 5) = 14.

• The digital root of a (also repeated digit sum) of a number is the single digit value obtained by an

iterative process of summing digits.

Highlight: This session deals with application based questions, which are of moderate difficulty level.

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SessionNumbers

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

Q1. A third standard teacher gave a simple multiplication exercise to the kids. But one kid reversed the

digits of both the numbers and carried out the multiplication and found that the product was exactly

the same as the one expected by the teacher. Only one of the following pairs of numbers will fit in

the description of the exercise. Which one is that?

(a) 14, 22 (b) 13, 62 (c) 19, 33 (d) 42, 28 (CAT 1991)

Q2. If 8 + 12 = 2, 7 + 14 = 3, then 10 + 18 = ?

(a) 10 (b) 4 (c) 6 (d) 18

(CAT 1991)

Q3. Let a, b, c be distinct digits. Consider a two-digit number ‘ab’ and a three-digit number ‘ccb’, both

defined under the usual decimal number system, if 2(ab) ccb 300,= > then the value of b is

(a) 1 (b) 0 (c) 5 (d) 6 (CAT 1999)

Q4. Consider four-digit numbers for which the first two digits are equal and the last two digits are alsoequal. How many such numbers are perfect squares?(a) 3 (b) 2 (c) 4 (d) 0 (e) 1

(CAT 2007)

Q5. Suppose, the seed of any positive integer n is defined as follows:

seed(n) = n, if n < 10

= seed(s(n)), otherwise,

where s(n) indicates the sum of digits of n. For example,

seed(7) = 7, seed(248) = seed(2 + 4 + 8) = seed(14) = seed(1 + 4) = seed(5) = 5 etc.

How many positive integers n, such that n < 500, will have seed (n) = 9?

(a) 39 (b) 72 (c) 81 (d) 108 (e) 55

(CAT 2008)

Q6. Let X be a four-digit positive integer such that the unit digit of X is prime and the product of all digitsof X is also prime. How many such integers are possible?(a) 4 (b) 8 (c) 12 (d) 24 (e) None of these

(XAT 2010)