Help with this algebra problem

Question

1. Davis says that adding a two-digit number to the two-digit number formed by reversing the digits of the original number results in a sum of 65.
2. a. Avery says that’s impossible. Is it impossible? Explain why or why not.
3. b. Avery offers a counter challenge to Davis. She says by following his addition scheme she got a sum of 154.
4. Find the two numbers Avery used.
5. c. Describe a method (other than addition!) of finding other sums that fit this pattern!

Answer 1

let ab and ba be the numbers

10a+b+10b+a=65

11a+11b=65

11(a+b)=65

65 is not divisible by 11 so a+b cannot be an integer

10a+b+10b+a=154

11a+11b=154

11(a+b)=154

a+b=154/11

a+b=14

a=5, b=9 gives 59 and 95, 59+95=154

a=6, b=8 gives 68 and 86, 68+86=154

a=7, b=7 gives 77 and 77, 77+77=154