factors

How many factors of 7¹⁴ x 8⁷ x 10⁵ are a multiple of 160?

Answer:

Expressing  7¹⁴ x 8⁷ x 10⁵ in terms of prime factors we have

2⁵ x 7¹⁴ x 2²¹ x 5⁵ = 2²⁶ x 7¹⁴ x 5⁵

Expressing 160 in terms of prime factors we have

2 x 2 x 2 x 2 x 2 x 5 = 2⁵ x 5

We will find the quotient when 2²⁶ x 7¹⁴ x 5⁵ is divided by 2⁵ x 5

We get 2²¹ x 7¹⁴ x 5⁴

To find the number of factors that are multiple of 160 we have to simple find the factors of 2²¹ x 7¹⁴ x 5⁴

Which is (21 + 1)(14 + 1)(4 + 1) = 22 x 15 x 5 = 1650

Answer: 1650