Question

in base ten, the two-digit prime N is 45 more than the number formed by reversing the digits of N, find all the possible values of N.

Answer 1

Let
`N=10x+y`, so that
`x` is the digits in the tens place and
`y` is the digit in the ones place. (Clearly
`x\\eq0`.)


`10x+y=45+10y+x\implies 9x-9y=45\implies x-y=5`

There are five possible two digits integers that satisfy this relation:


`x=5,y=0\implies N=50`

`x=6,y=1\implies N=61`

`x=7,y=2\implies N=72`

`x=8,y=3\implies N=83`

`x=9,y=4\implies N=94`

But the first, third, and fifth candidates are even, so they are not prime. The remaining are prime, however, so
`N=61` or
`N=83`.