CAT 2023 Slot 1 QA Question 2

Multiple choice (+3 / −1) · Number Systems · Factorisation · Try it, then check the answer and solution below.

Let

n

be the least positive integer such that 168 is a factor of

1134^n

. If m is the least positive integer such that

1134^{n}

. is a factor of

168^m , then

m + n$ equals

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Official Convenor Answer Key: C

Answer and solution

Answer: C) $15$

To solve the problem, we need to determine the least positive integers

n

and

m

such that 168 is a factor of

1134^n

and

1134^n

is a factor of

168^m , respectively. Then, we find

m + n$.

First, we factorize 168 and 1134:

168 = 2^3 \times 3^1 \times 7^1

1134 = 2^1 \times 3^4 \times 7^1

Finding

n

We need the smallest

n

such that 168 divides

1134^n

. This means the exponents of the prime factors in 168 must be less than or equal to those in

1134^n

For 2:

3 \leq n

For 3:

1 \leq 4n

(automatically satisfied forn ≥ 1)

For 7:

1 \leq n

The most restrictive condition is

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