For all natural numbers k m and n if km and kn then km n if

For all natural numbers k, m, and n: if k|m and k|n then k|(m + n), if k|(m + n) then k|m and k|n, if Aim and k|n then k|mn, if k|m or k|n then k|mn, if k|m and k|n,

Solution

1. In first we can see m and n both are the factors of k. But it is not necessary that (m+n) will be the factor of k. For instance, k=6, m=3, n=2.

6%2=0, 6%3=0 (Remainder is 0 in both cases that means m and n are the factor of k)

but 6%(2+3) = 6%5 = 1.(Remainder is 1 in both cases that means m+n is not the factor of k)

Therefore we disprove it.

2. Same reason as 1. So we disprove it. Hence k|(m+n) may or may not be a natural no.

3. m is the factor of k and n is also the factor of the k. Therefore we can say that

k = m*n*x.......(i)

where x is an another natural number which is also the factor of k.

from (i), we can conclude that m*n is also the factor of k. That means k|m*n is also a natural number.

4. m is factor of k or n is the factor of k. But they both may or may not be the factor of k.

Therefore we can say

k=m*x

or

k=n*x

or

k=m*n*x

x is any natural number.

We are not sure about k|m*n is natural number or not. Therefore we disprove it.

5. Same reason as 3. That implies k|m and k|n are both natural numbers.