Question

Let n be the smallest composite number such that it can be written as the product of two positive integers that differ by 10. How many distinct prime factors does n have?

Answer 1

2

Explanation:

Solution:

The smallest pair of numbers that differ by 10 is 1 and 11; however, they multiply to 11 which is not composite. We try the next pair up: 2 and 12, which multiply to 24 which is composite, so
`$n=24$`.
`$24$` factors as
`$2^3 \cdot 3$`, so it has
`$\boxed{2}$` prime factors: 2 and 3.

Hope this helped! :)

Answer 2

Answer: n has two distinct prime factors 2 and 3.

Explanation:

As given the smallest composite number such that it can be written as the product of two positive integers that differ by 10.

i.e. if first number be x then the other number be x+10 and n=x(x+10), where n must be composite number.

So after putting the values of x by positive integer

1) x=0 then n=0(10+0)=0...which is not a composite number.

2) x=1 then n=1(1+10)=11..... which is not a composite number.

3)x=2 then n=2(2+10)=24......which is a composite number.

so our n=24 and we know that 24=2×2×2×3

Therefore its distinct prime factors are 2 and 3.