A positive integer n can be expressed in the form p^3 q^2, where p and q are distinct prim… — GMAT

A positive integer $n$ can be expressed in the form $p^3 q^2$ , where $p$ and $q$ are distinct prime numbers. If $n$ is divisible by 45 and is a factor of 4,050, what is the value of $n$ ?

Answer: 675

Answer

675

Prime factorizing

45 = 3^2 \cdot 5

and

4,050 = 2 \cdot 3^4 \cdot 5^2

shows that

n

must be composed of the prime factors 3 and 5. The two candidate values for

p^3 q^2

are

3^3 \cdot 5^2 = 675

and

5^3 \cdot 3^2 = 1,125

. Both are multiples of 45, but only 675 is a factor of 4,050 because its power of 5 does not exceed

5^2

Step-by-Step Solution

Determine the prime factorizations of 45 and 4,050

45 = 3^2 \cdot 5^1

and

4,050 = 2^1 \cdot 3^4 \cdot 5^2

Decomposing the given numbers into prime factorizations determines the prime building blocks for

n

Identify the distinct prime factors

p

and

q

The primes

p

and

q

must be 3 and 5

Since

n

is divisible by 45, its prime factorization must contain at least

3^2

and

5^1

. Since

n = p^3 q^2

has exactly two distinct prime factors, those factors must be 3 and 5.

Evaluate both possible assignments for

p

and

q

n = 3^3 \cdot 5^2 = 675

n = 5^3 \cdot 3^2 = 1,125

Assigning 3 to

p

and 5 to

q

yields 675, while assigning 5 to

p

and 3 to

q

yields 1,125.

Filter using the constraint that

n

is a factor of 4,050

675 is a factor of 4,050; 1,125 is not a factor of 4,050

For a number to be a factor of

4,050 = 2^1 \cdot 3^4 \cdot 5^2

, the exponent of prime 5 cannot exceed 2. In 1,125, the exponent of 5 is 3, which makes 1,125 invalid.

Key Concept

Prime Factorization, Divisibility Rules, and Factors of Integers

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