## Divisor and Multiple Basic Math Problem

• We are trying to make a square number by multiplying the smallest natural number to $240$. Which natural number should be multiplied?
##### solution :

Prime factorization $240=2^4\times&space;3\times&space;5$ . To make a square number all indexes are even number. So we must multiplying $3$ and $5$$3\times&space;5$.

##### Answer : $15$

• What is the number that has the most factor among these numbers?

(1) $9$     (2) $12$     (3) $28$     (4) $72$     (5) $135$

##### solution :

To find the number of factor, 1) Prime factorization 2) all index number $+&space;\:1$ 3) and plus that new numbers.

(1) $9=3^2$  index $2+1=3$ so number of factor of $9$ is $3$

(2) $12=2^2\times&space;3^1$ index $\left&space;(&space;2+1&space;\right&space;)&space;\times&space;\left&space;(&space;1+1&space;\right&space;)=6$ , number of factor of $12$ is $6$

(3) $28=2^2\times&space;7^1$ index $\left&space;(&space;2+1&space;\right&space;)&space;\times&space;\left&space;(&space;1+1&space;\right&space;)=6$ , number of factor of $28$ is $6$

(4) $72=2^3\times&space;3^2$  index $\left&space;(&space;3+1&space;\right&space;)\times&space;\left&space;(&space;2+1&space;\right&space;)=12$ , number of factor of $72$ is $12$

(5) $135=3^3\times&space;5^1$ index $\left&space;(&space;3+1&space;\right&space;)\times&space;\left&space;(&space;1+1&space;\right&space;)=8$,  number of factor of $135$ is $8$