Question

For a positive number N, if N² is completely divisible by 24, then which of the following is the greatest number by which N must be completely divisible?

• A. 6
• B. 12
• C. 18
• D. 24

Answer 1

The Correct Option is B

Step 1: Understanding the Divisibility Condition 

We are given that \( N^2 \) is divisible by 24. Since:

\[ 24 = 2^3 \times 3 \]

This means that \( N^2 \) must be divisible by both 8 and 3. Therefore, \( N \) must have at least half the prime powers of 24 in its prime factorization.

Step 2: Determining the Smallest Integer for \( N \)

For \( N^2 \) to be divisible by 24, \( N \) must be divisible by the square root of 24:

\[ \sqrt{24} = \sqrt{2^3 \times 3} = 2^{\frac{3}{2}} \times 3^{\frac{1}{2}} \]

The smallest integer satisfying this condition is:

\[ N = 2^2 \times 3 = 12 \]

Step 3: Conclusion

Therefore, the greatest number by which \( N \) must be divisible is 12.