CAT 2024 Slot 2 QA Question 3

The question says: If m and n are natural numbers such that n > 1, and m^n equals 2^25 multiplied by 3^40, then m - n equals which of the options given? The options are A) 209937, B) 209947, C) 209942, D) 209932. The correct answer is supposed to be B, but We analyze why.

Alright, so first, let's break down what's given. We have m^n = 2^25 × 3^40. Both m and n are natural numbers, and n is greater than 1. We need to find m - n.

Hmm, okay. So, m raised to the power of n equals this product of powers of 2 and 3. Since m and n are natural numbers, m must be an integer, and n must be an integer greater than 1.

when dealing with exponents, especially prime factors, it's useful to express m in terms of its prime factors. So, let's assume that m can be written as 2^a × 3^b, where a and b are non-negative integers. That makes sense because 2 and 3 are primes, and any natural number can be broken down into prime factors.

So, if m = 2^a × 3^b, then m^n would be (2^a × 3^b)^n, which is equal to 2^(an) × 3^(bn). According to the problem, this equals 2^25 × 3^40.

Therefore, we can set up equations for the exponents:

an = 25 bn = 40

So, we have two equations:

Our goal is to find m and n, so we need to find a, b, and n. Since n is a common factor in both equations, it must be a common divisor of 25 and 40.

Let me list the divisors of 25 and 40 to find possible values of n.

Divisors of 25: 1, 5, 25 Divisors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Since n > 1, the common divisors greater than 1 are 5.

So, n must be 5.

If n = 5, then from the first equation:

a*5 = 25 => a = 25/5 = 5

From the second equation:

b*5 = 40 => b = 40/5 = 8

So, a = 5 and b = 8. Therefore, m = 2^5 × 3^8.

Compute m:

First, 2^5 is 32.

3^8 is a bit bigger. Computing that step by step.

3^2 = 9 3^4 = (3^2)^2 = 9^2 = 81 3^8 = (3^4)^2 = 81^2

Calculating 81^2: 80^2 is 6400, and 81^2 is 6561.

So, m = 32 × 6561.

Compute that:

32 × 6561. Hmm, 32 × 6000 = 192,000 32 × 500 = 16,000 32 × 61 = 1,952

Wait, that might not be the best way. Alternatively, 32 × 6561 can be calculated as:

32 × 6561 = (30 + 2) × 6561 = 30×6561 + 2×6561

30×6561: 6561 × 10 = 65,610; so 65,610 × 3 = 196,830

2×6561 = 13,122

Adding them together: 196,830 + 13,122 = 209,952

So, m = 209,952.

Wait, but looking at the options, the numbers are around 209,937 to 209,947. Hmm, 209,952 is close but not exactly matching. Did I make a mistake in calculation?

Wait, m = 2^5 × 3^8.

2^5 is definitely 32.

3^8: Computing it again.

3^1 = 3 3^2 = 9 3^3 = 27 3^4 = 81 3^5 = 243 3^6 = 729 3^7 = 2,187 3^8 = 6,561

Yes, that's correct. So, 3^8 is 6,561.

Then, 32 × 6,561.

Compute 32 × 6,561 step by step.

First, 30 × 6,561 = 196,830

Then, 2 × 6,561 = 13,122

Adding them: 196,830 + 13,122 = 209,952

Hmm, that's correct. So, m = 209,952.

But the options are:

Wait, none of these options is 209,952. Did I do something wrong?

Maybe I made a mistake in assuming n is 5.

Earlier, I thought the common divisors of 25 and 40 greater than 1 are only 5. But let me confirm.

25: 1, 5, 25

40: 1, 2, 4, 5, 8, 10, 20, 40

So,