4
$\begingroup$

If we have integers $h$, $i$, $j$, and $k$, would it be true to say that $\gcd(h,i)\gcd(j,k)|\gcd(hi,jk)$? If so, how can we prove it?

  • 1
    Maybe you want $\gcd(h,j)\gcd(i,k)|\gcd(hi,jk)$?2011-10-14

3 Answers 3

5

No, consider $h=i=2$ and $j=k=3$, then $\gcd(h,i)\gcd(j,k)=\gcd(2,2)\gcd(3,3)=2\cdot 3=6\nmid 1=\gcd(4,9).$

2

By distributivity $\rm\ (h,i)\:(j,k) = (hj,hk,ij,ik)\ $ which divides $\rm\ hj, ik\ $ so also $\rm\:(hj,ik)\:.\:$ Presumably that's what was intended.

-2

This statement is false, You will see it whenever you take $h,i,j,k$ such that the products $hi$ and $jk$ are coprime but the numbers $h,i$ and $j,k$ are not pairwise coprime.

Illustrations:

Select $h=2, i=4$ and $j=3,k=9$, you will see that $6\not| 1$.

Select $h=5,i=15$ and $j=2,k=4$, you will again see that $10\not|1$.

  • 0
    Bhai, tere number pe call nhi lag raha hai, time mile to call back kariyo... KVPY ka result aa gaya hai.2017-12-22