If we take a natural number N the number formed by subtracti

If we take a natural number N, the number formed by subtracting double the last digit of N from the number formed by the rest of the digits is divisible by 7 if and only if N is divisible by seven. (Example: 721 is divisible by 7 because 72-2(1)=70 is divisible by 7.)

Solution

Let last digit of N be x

Let number formed by subtracting double of last digit of N from number formed by rest of the digits be divisible by 7

ie (N-x)/10-2x=(N-21x)/10=0 mod 7

Hence, N-21x=0 mod 7

N=21x=0 mod 7

Because,21 is multiple of 7

Hence, N=0 mod 7

Now let N be a multiple of 7

(N-21x)/10=N-21x=N =0 mod 7

Hence proved