Since, The integer 𝑘 is equal to \(𝑚^2\) for some integer 𝑚.
A perfect square is a number that can be expressed as the product of two equal integers. This is why 𝑘 must be a perfect square.

If we take 𝑘 as \((2^2)(3^2)(5^2) = 900\), it is not divisible by 24
\(\frac{900}{24} = 37.5\) not an integer

To solve this, let's find the smallest k such that k = m^2 and k is divisible by both 20 and 24. Start by finding the least common multiple (LCM) of 20 and 24, which will be the smallest k that satisfies the divisibility requirements.

1. Factorize 20 and 24:
20 = 2^2 * 5
24 = 2^3 * 3

2. Find the LCM:
- The LCM will include each prime factor raised to its highest power across both factorizations:
- LCM(20, 24) = 2^3 * 3 * 5 = 120

3. Since k = m^2 and m must be an integer, k must be a perfect square. The smallest perfect square that is at least 120 and divisible by both 20 and 24 is 3600:

For k = m^2 and k to be a perfect square, every prime factor in its factorization must appear to an even power. From the step above, the minimal factorization needed is 2^3 * 3 * 5 . To make each of these powers even, adjust them to the next even power (since the square of any integer will have even powers for all primes):
Increase 2^3 to 2^4 (next even power)
Increase 3 to 3^2
Increase 5 to 5^2

4. Multiply these adjusted factors:

2^4 * 3^2 * 5^2 = 16 * 9 * 25 = 3600.

If you did not know the rule of exponents and perfect squares, or another shortcut, at step 3, you may have spent a long time crunching tough numbers. Or even aborted the mission at that point. It’s a hard question.

You need to know these rules. The Ultimate GRE Cheat Sheet has excellent chapters on exponent rules and divisibility. I strongly recommend you get a copy.

With 160 pages of rules and patterns, the cheat sheet is much more comprehensive than the lists of formulas and rules that you will see elsewhere.

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