Answer:
- 120 = 2³ · 3 · 5
- 315 · 35 = 105²; n = 35
Step-by-step 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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