Let [tex]N=10x+y[/tex], so that [tex]x[/tex] is the digits in the tens place and [tex]y[/tex] is the digit in the ones place. (Clearly [tex]x\neq0[/tex].)
[tex]10x+y=45+10y+x\implies 9x-9y=45\implies x-y=5[/tex]
There are five possible two digits integers that satisfy this relation:
[tex]x=5,y=0\implies N=50[/tex]
[tex]x=6,y=1\implies N=61[/tex]
[tex]x=7,y=2\implies N=72[/tex]
[tex]x=8,y=3\implies N=83[/tex]
[tex]x=9,y=4\implies N=94[/tex]
But the first, third, and fifth candidates are even, so they are not prime. The remaining are prime, however, so [tex]N=61[/tex] or [tex]N=83[/tex].
