Maths olympiad lesson: last digits of powers, cycles and remainders

Last digits: practice

Show your working: the cycle, the remainder, the place in the cycle. The last questions are proofs: check every possible last digit.

Diagnostic question: choose and explain

1. What is the last digit of 3¹⁰⁰?
   1. 3
   2. 1
   3. 9
   4. 0

Practice questions

1. What is the last digit of 2¹⁰?
2. The last digits of 4¹, 4², 4³, 4⁴ are ___, ___, ___, ___: a cycle of length ___
3. What is the last digit of 9²⁰²⁷?
4. What is the last digit of 13²⁰²⁷?
5. Why must the last digits of 7, 7², 7³, … repeat?
   1. Because 7 is a prime number
   2. There are only ten last digits, and each one fixes the next
   3. Because 7⁴ = 2401 ends in 1
   4. They needn't: we only checked the first twelve
6. What is the remainder when 7²⁰²⁶ is divided by 5?
7. Prove that no square number ends in 2, 3, 7 or 8.
8. What is the last digit of 2²⁰²⁶ + 3²⁰²⁶ + 4²⁰²⁶?
9. Stretch: what is the last digit of 1 + 3 + 3² + 3³ + … + 3²⁰²⁵?
10. Stretch: prove that n⁵ − n ends in 0 for every whole number n.

Support: Write the cycle out, number its places 1, 2, 3, 4, and say which place remainder 0 is. For the proofs: list every possible last digit, 0 to 9, and check each one.

Last digits: exit

A digit, then a sentence or two.

1. What is the last digit of 8²⁰²⁶?
2. Say how the cycle tells you the last digit of 8²⁰²⁶.

Exit check

One last digit, with the reason, shows who can use a cycle.

What is the last digit of 8²⁰²⁶?

Last digits: cycles and remainders

For each, write the cycle, the remainder and the place in the cycle. Question 5 is a proof; question 7 is a challenge.