Question

The digits of a two-digit number differ by 2. If the digits are interchanged and the resulting number is added to the original number, we get 110. What can be the original number?

Answer 1

`x` - tens digit

`y` - units digit

`10x+y` - the original number

`10y+x` - the number with interchanged digits

`|x-y|=2\\10x+y+10y+x=110\\\\x-y=2 \vee x-y=-2\\11x+11y=110\\\\x=y+2 \vee x=y-2\\x+y=10\\\\y+2+y=10 \vee y-2+y=10\\2y=8 \vee 2x=12\\y=4 \vee y=6\\\\x+4=10 \vee x+6=10\\x=6 \vee x=4`

`10x+y=10\cdot 6+4 \vee 10x+y=10\cdot4+6\\10x+y=64 \vee 10x+y=46`

Therefore, the original number can be either 46 or 64.