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The digits of a five-digit number are MNOPQ. Prove that MNOPQ is divisible by 7 if and only of the number MNOP - 2 x Q is divisible by 7.

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- May 8th 2010, 06:52 AMmathisgoodNumber Theory
Help on this...

The digits of a five-digit number are MNOPQ. Prove that MNOPQ is divisible by 7 if and only of the number MNOP - 2 x Q is divisible by 7.

Thanks! - May 8th 2010, 07:05 AMAnonymous1
- May 8th 2010, 08:48 AMSoroban
Hello, mathisgood!

Quote:

The digits of a five-digit number are

Prove that is divisible by 7 if and only of the number is divisible by 7.

[1] Given: . is divisible by 7.

. . .Prove: . is divisible by 7.

We have: .

Subtract

Divide by 10: .

Since is an integer, then is a multiple of 10: .

And we have: .

Therefore: . is divisible by 7.

[2] Given: . is divisible by 7.

. . .Prove: . is divisible by 7.

We have: .

. . . Then: .

Multiply by 10: .

. . Add

And we have: .

Therefore: . is divisible by 7.