Divisibility By 7 and 11

Divisibility By 7 and 11

Any number can be written in the form of 10x+y, where y is the unit digit and x is the remaining part.

Example: 3221=322×10+1                             x=322 and y=1

                  32479=3247×10+9                        x=3247 and y=9

Divisibility by 11:

If 10x+y is divisible by 11, then -10x-y is divisible by 11

And if 11x is divisible by 11, then their sum 11x+(-10x-y)=x-y is divisible by 11

Example: 1089: x=108 and y=9                   x-y=108-9=99 is divisible by 11 then 1089 is divisible by 11

Example: 722238: x=72223 and y=8         x-y=72223-8=72215

                     72215: x=7221 and y-5                x-y=7221-5=7216

                     7216: x=721 and y=6                   x-y=721-6=715

                     715: x=71 and y=5                        x-y=71-5=66

66 is divisible by 11 then 722238 is divisible by11

Divisibility by 7:

If 10x+y is divisible by 7 then -10x-y is divisible by 7 and 7x+7y is divisible by 7, then their sum is divisible by 7

-10x-y+7x+7y is divisible by 7

-3x+6y is divisible by 7

-3(x-2y) is divisible by 7

-3 is not divisible by 7 so x-2y is divisible by 7

Example:14924: x=1492 and y=4                               x-2y=1492-8=1484

                    1484: x=148 and y=4                                   x-2y=148-8=140

                    140: x=14 and y=0                                        x-2y=14-0=14

14 is divisible by 7, then 14924 is divisible by 7