GMAT Practice Question: If n is a positive integer and n^{2}...
Question
If n is a positive integer and n^{2} is divisible by 72 , then the largest positive integer that must divide n is
- 6
- 12
- 24
- 36
- 48
Topics: number theory, divisibility, prime factorization
Solution
Step 1
Factor 72 into its prime factors
72 = 2^{3} \times 3^{2}
Step 2
Determine the minimum exponent of 2 in n so that n^{2} includes at least a factor of 2^{3}
\text{If }n\text{ contains }2^{k},\text{ then }n^{2}\text{ contains }2^{2k},\text{ and we need }2k \ge 3,\text{ so }k = 2
Step 3
Determine the minimum exponent of 3 in n so that n^{2} includes at least a factor of 3^{2}
\text{If }n\text{ contains }3^{m},\text{ then }n^{2}\text{ contains }3^{2m},\text{ and we need }2m \ge 2,\text{ so }m = 1
Step 4
Combine the required prime factors to find the largest integer that must divide n
2^{2} \times 3^{1} = 12
Answer
12 - Option B
