Number Theory · Topic 8 of 9

🔚 Unit Digit of Powers

1 exam question type, fully solved

“What is the last digit of 7⁴⁵?” You can never compute that number — but you never need to. Two simple rules handle every such question.

The two rules — learn these properly

Rule 1 — Only the last digit matters. For a product, multiply the last digits and keep the last digit of the result (so 14 × 27 ends in 4 × 7 = 28 → 8). For a sum or difference, add or subtract the last digits; if the result is negative, add 10.

Rule 2 — The last digit of a power repeats in a cycle.

• Digits 0, 1, 5, 6 → the last digit never changes (cycle length 1).

• Digits 4, 9 → repeat every 2 powers.

• Digits 2, 3, 7, 8 → repeat every 4 powers.

How to use it: take the last digit of the base, find its cycle, then divide the power by the cycle length and use the remainder to pick the term. If the remainder is 0, use the last term of the cycle.

Worked example — 7⁴⁵: base ends in 7, cycle 7, 9, 3, 1 (length 4). 45 ÷ 4 leaves remainder 1 → use the 1st term → last digit 7.

🎡Cyclicity WheelLast digit of any power

Pick the base — only its last digit matters

Power (exponent)

7 repeats its last digit every 4 powers: [ 7, 9, 3, 1 ]. 45 ÷ 4 leaves remainder 1 → use position 1 of the cycle.

Last digit of 7⁴⁵ = 7

Last digit of 2³⁰: cycle of 2 is 2, 4, 8, 6. 30 ÷ 4 leaves 2, so use 2² = 4 → last digit .

Q1The unit digit of 7⁴⁵ is:

Q2The unit digit of 6¹⁰⁰ is:

Q3The unit digit of 2⁴ × 3³ is:

📚

Real exam questions — Unit Digit

1 question types · 3 solved examples from real SSC papers

You can never compute 7⁴⁵, but you never need to — only the repeating cycle of the last digit matters.

How to solve this type

Only the unit digit of the base matters, never the full number. Digits 0, 1, 5 and 6 always keep the same unit digit; 4 and 9 repeat every 2 powers; 2, 3, 7 and 8 repeat every 4 powers. For a cycle of 4, divide the exponent by 4 and take the remainder; if the remainder is 0, use the 4th term of the cycle (not the 0th). For products, find each term's unit digit and multiply them. For sums or differences, combine the unit digits, and if the result is negative add 10.

Find the unit digit of (164)¹⁶⁹ + (333)³³⁷ − (727)⁷²⁶.

A5B9C8D7

Find the unit digit of 11³⁴ × 22¹⁵ × 33⁴⁶.

A4B8C6D2

Find the unit digit of 3 + 3² + 3³ + … + 3⁸.

A2B4C6D0