Study guides › Level 8 · Algebra 2

The last digit of a big power

Level 8 · Algebra 2 · for questions like “What is the last digit of 7⁶³?”

💡 The idea

The last digits of powers repeat in a loop. Find the loop, then find where the power lands in it.

🪜 The steps

1. Write the last digits of the first few powers: 7, 9, 3, 1, 7, 9, …
2. The loop length here is 4.
3. Divide the power by the loop length: 63 ÷ 4 leaves 3, so take the 3rd digit in the loop.

✏️ Worked examples

1. 7⁶³: the loop is 7, 9, 3, 1. 63 leaves 3, so the last digit is 3.

2. 2⁴⁰: the loop is 2, 4, 8, 6. 40 leaves 0, which means the 4th digit: 6.

⚠️ Watch out

• Remainder 0 means the last digit in the loop, not the first.

🎯 Try these

1. 3²⁵
2. 8¹⁰
3. 9⁵⁰

Answers: 1) 3 2) 4 3) 1

This guide is also in the 10MathQ app: tap “📖 Study guide” after a question.