Question

A) write 120 as product of its prime factors.

B) 315=3^2×5×7

Use this formation to find the smallest integer value of n, such that 315n is a square number.

Answer 1

• 120 = 2³ · 3 · 5
• 315 · 35 = 105²; n = 35

Explanation:

You want the prime factors of 120, and the smallest positive multiplier of 315 that makes the product a perfect square.

A) Factors

You know that 120 = 12·10, and that each of these factors can be factored further:

120 = (4·3) · (2·5) = 2³ · 3 · 5

B) Square

If a number is a perfect square, each of its prime factors has an even exponent. The factors 5 and 7 of 315 each have an exponent of 1, so the smallest n such that 315n is a square will be ...

n = 5·7 = 35

The product 315·35 is (3²·5·7)(5·7) = 3²·5²·7² = 105²

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