Number System & Simplification • Topic 4 of 6

Factors & Unit Digit

The number of factors of N = p^a x q^b x r^c is (a+1)(b+1)(c+1). The unit digit of a power follows a cycle of length at most 4: 2 cycles 2,4,8,6; 3 cycles 3,9,7,1; 7 cycles 7,9,3,1. To find the unit digit of a^n, reduce n using the cycle (effectively mod 4) and read the matching digit. Digits 0,1,5,6 always repeat themselves.

✅ Solved examples

1. How many factors does 60 have?
60 = 2^2 x 3 x 5. Factors = (2+1)(1+1)(1+1) = 3 x 2 x 2 = 12.
2. Find the unit digit of 7^53.
7 cycles 7,9,3,1 (length 4). 53 mod 4 = 1, so unit digit = first in cycle = 7.
3. Unit digit of 2^40?
2 cycles 2,4,8,6. 40 mod 4 = 0 -> last in cycle = 6.
4. How many odd factors does 90 have?
90 = 2 x 3^2 x 5. Drop the 2: 3^2 x 5 has (2+1)(1+1) = 6 odd factors.

✏️ Practice — try these, take hints as needed

1. Number of factors of 84?
84 = 2^2 x 3 x 7.
(2+1)(1+1)(1+1).
—
12
2. Unit digit of 3^24?
Cycle 3,9,7,1.
24 mod 4 = 0.
Last in cycle.
1
3. Unit digit of 4^17?
4 cycles 4,6.
Odd power -> 4.
—
4
4. How many even factors does 72 have?
72 = 2^3 x 3^2; total (3+1)(2+1) = 12.
Odd factors = 3^2 -> 3.
12 - 3.
9
5. Unit digit of 8^25?
8 cycles 8,4,2,6.
25 mod 4 = 1.
First in cycle.
8

📝 Topic test — 8 questions

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