KA-0037Last-digit behavior of products and powers
5 points, difficulty 3 of 3Level 5–6about 120sWhat is the units digit of 7 multiplied by itself 2026 times, that is 7 to the power 2026?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
List the last digits of the first few powers.
2The strategy
The last digit of a product depends only on the last digits multiplied. Work out the last digits of the first few powers until they repeat, then use the remainder to find the position.
3The full solution
The last digits go 7, 9, 3, 1 and then repeat every 4. 2026 divided by 4 is 506 remainder 2, so the answer is the second in the cycle, which is 9.
Solution
The reliable way
The last digit of a product depends only on the last digits of the factors, so build the cycle: 7 to the power 1 ends in 7, then 49 ends in 9, then 343 ends in 3, then 2401 ends in 1, then it starts again at 7. The cycle has length 4. Divide 2026 by 4 to get 506 remainder 2, so the answer sits at position 2 of the cycle, which is 9. The transferable idea: find the cycle, then use the remainder as a position within it.
The elegant way
Powers of 7 repeat their last digit every 4 steps, and 2026 is 2 more than a multiple of 4, so it ends like 7 squared - in 9.
Why this is on the test: It looks impossible without a calculator and takes twenty seconds with the cycle, which is exactly the kind of trade a 5-point question rewards.