What is the smallest positive integer n such that 180 \times n is a perfect square and 450… — GMAT

What is the smallest positive integer $n$ such that $180 \times n$ is a perfect square and $450 \times n$ is a perfect cube?

Answer: 1620

Answer

The smallest positive integer

n

is 1620.

By writing 180 as

2^2 \times 3^2 \times 5^1

and 450 as

2^1 \times 3^2 \times 5^2

, we can analyze the exponents needed for

n = 2^a \times 3^b \times 5^c

. To make

180n

a perfect square,

a

and

b

must be even and

c

must be odd. To make

450n

a perfect cube,

1+a

2+b

, and

2+c

must be multiples of 3. Finding the minimal non-negative integers that satisfy both conditions yields

a = 2

b = 4

, and

c = 1

. Thus,

n = 2^2 \times 3^4 \times 5^1 = 1620

Step-by-Step Solution

Express 180 and 450 in terms of their prime factorizations.

180 = 2^2 \times 3^2 \times 5^1

and

450 = 2^1 \times 3^2 \times 5^2

Prime factorization exposes the exponent requirements for perfect powers.

Determine the constraints on exponents of

n = 2^a \times 3^b \times 5^c

For

180n

to be a square,

a

must be even,

b

must be even, and

c

must be odd. For

450n

to be a cube,

1+a

2+b

, and

2+c

must be multiples of 3.

A number is a perfect square if all prime exponents are even, and a perfect cube if all prime exponents are multiples of 3.

Find the minimal values for

a, b, c

a = 2, b = 4, c = 1

a=2

is even and makes

1+2=3

;

b=4

is even and makes

2+4=6

;

c=1

is odd and makes

2+1=3

Compute

n

n = 2^2 \times 3^4 \times 5^1 = 1620

Multiplying the prime powers together yields the smallest integer

n

Key Concept

Prime Factorization and Exponent Rules for Perfect Powers

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