35m + 8 \geq 10 \Rightarrow m \geq 0, \quad 35m + 8 \leq 99 \Rightarrow m \leq 2
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- 7k + 1 \equiv 3 \pmod{5} \Rightarrow 7k \equiv 2 \pmod{5} \Rightarrow 2k \equiv 2 \pmod{5} \Rightarrow k \equiv 1 \pmod{5}
- So $ k = 5m + 1 $, and $ x = 7(5m + 1) + 1 = 35m + 8 $.
- We now find a two-digit value of $ x $:
- Try $ m = 0 $: $ x = 8 $ (not two-digit)
- Try $ m = 1 $: $ x = 43 $
- Try $ m = 2 $: $ x = 78 $
