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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

#1
+32301
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Jul 30, 2015

#1
+32301
+21

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Alan Jul 30, 2015
#2
+113752
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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
+32301
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"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
+113752
+1

Thankyou Alan

Aug 1, 2015