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# Let \$x\$ and \$y\$ be integers. Show that \$9x + 5y\$ is divisible by 19 if and only if \$x + 9y\$ is divisible by 19.

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Let \$x\$ and \$y\$ be integers. Show that \$9x + 5y\$ is divisible by 19 if and only if \$x + 9y\$ is divisible by 19.

Jul 30, 2015

### Best Answer

#1
+21 .

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Jul 30, 2015

### 4+0 Answers

#1
+21
Best Answer .

Alan Jul 30, 2015
#2
+11

I really like this solution Alan, thank you.

For the last bit you just have to show that 9m/19 is not an integer for  0<m<19

can't you just say that 9 and 19 are relatively prime so 9m/19 will only be an integer if m is a multiple of 19.  Since m cannot be a multiple of 19, 9m/19 is not an integer.

That is, 9m is not divisable by 19.

Jul 31, 2015
#3
+5

"For the last bit you just have to show that 9m/19 is not an integer for  0<m<19 can't you just say that 9 and 19 are relatively prime so 9m/19 will only be an integer if m is a multiple of 19.  Since m cannot be a multiple of 19, 9m/19 is not an integer. That is, 9m is not divisable by 19"

Yes.  Much neater Melody.

Aug 1, 2015
#4
+1

Thankyou Alan Aug 1, 2015