Show that if (7x+2y-5z) is divisible by 11, (3x-7y+12z) is also divisible by 11.

If x, y and z are whole numbers, and (7x+2y-5z) is divisible by 11, show that (3x-7y+12z) is also divisible by 11.

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Since 7x+2y-5z is divisible by 11, we have

`7x+2y-5z equiv 0 (mod 11)`

Now consider

`3x-7y+12z` also mod 11 and add appropriate multiples of 11

`equiv 14x+4y-10z` after adding 11x and 11y and subtracting 22z

`equiv 2(7x+2y-5z)`

`equiv 2(0)`

`equiv 0`

**Therefore 3x-7y+12z is a multiple of 11.**

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