Prove that (43^43) - (17^17) is divisible by 10

To be a multiple of 10, you're only concerned about tue units (ones) place. 

For 43^43, the units place is the result of 3^43. Powers of 3 go in the cycle: 3 9 7 1 3 9 7 1 3 ... (that is, 3, 9, 27, 81, 243, . . .). Taken 43 times, it goes through the cycle 3 9 7 1 ten full times, so 43^40 has 1 in the units place.  Another three times (3 9 7) shows that 3^43 ends in 7. 

Similarly. for 17^17. the units place goes through the cycle 7 9 3 1 7 9 3 1 7 . . .  Taken 17 times, it goes through the cycle 7 9 3 1 four full times, so 17^16 has 1 in the units place. Another one time shows that 17^17 ends in 7. 

Since both numbers, 43^43 and 17^17, have 7 in the units place, the difference, 43^43 - 17^17, ends in 7-7, or 0. Any number with 0 in the units place is, by definition, a multiple of 10. 

QED

((43 ^ 43) - (17 ^ 17)) / 10 = 1734377336703030 E+69

or

1,734,377,336,703,030,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000.00000