MathLabs

Problem 2

Show that (36a+b)(a+36b)(36a+b)(a+36b) cannot be a power of 22 for any positive integers aa and bb.
Step 2 of 4: Force divisibility by four
36a+b>36,a+36b>3636a+b>36,\qquad a+36b>36
Detailed analysis

Both factors exceed 3636, so each power of 22 is divisible by 44. Reducing the first factor modulo 44 gives b≡0(mod4)b\equiv0\pmod4; reducing the second gives a≡0(mod4)a\equiv0\pmod4.