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Number Theory · Topic 4 of 9

🔑 Factors, Primes & Squares

4 exam question types, fully solved

A prime has exactly two factors: 1 and itself. Every other number is built by multiplying primes — and prime factorisation is the master key to this whole topic.

The method that solves most questions

Step 1 — write the number in prime form, e.g. 360 = 2³ × 3² × 5. Then every question becomes one formula:

• Number of factors = (3+1)(2+1)(1+1) = 24  (add 1 to each power, then multiply)

• Odd factors = ignore the power of 2, use the rest

• Even factors = total − odd

• Sum of factors = (1+2+4+8)(1+3+9)(1+5) = 15 × 13 × 6 = 1170

Is N prime? Test only prime divisors up to √N. For 97: √97 ≈ 9.8, so test 2, 3, 5, 7 — none divide it, so 97 is prime.

🌳Prime FactorizerSee how any number breaks down into prime factors
NumberDivide byResult
60÷ 230
30÷ 215
15÷ 35
5PRIME ✓
60 = 2² × 3 × 5
2235=60
The #1 exam formula — add 1 to each power
Number of factors = (2+1) × (1+1) × (1+1) = 12Sum of factors = (2³−1)/(2−1) × (3²−1)/(3−1) × (5²−1)/(5−1) = 7 × 4 × 6 = 168
🔲Factor ExplorerEnter a number to see all its factor pairs as rectangles
6 factors
1234612
1 × 12
2 × 6
3 × 4
🧮Sieve of EratosthenesStep through to discover all prime numbers up to 100
Step 1 of 6

Start with all numbers from 1 to 100. First, cross out 1 — it has only one factor, so it's neither prime nor composite.

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■ Prime✗ Composite□ Unknown
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Number of factors of 12 = 2² × 3 is (2+1)(1+1) = . Is 97 prime? .

📝Practice Questions

Q1How many factors does 72 have?

Q2Which number is a perfect square?

Q3How many odd factors does 90 have?

📚

Real exam questions — Factors & Primes

4 question types · 12 solved examples from real SSC papers

Prime factorisation is the master key — it unlocks counting factors, summing factors, perfect squares and cubes.

How to solve this type
A prime has exactly two factors: 1 and itself. To test a number N, check division only by primes up to √N.
The only even prime is 2; every other prime is odd.
Use digit-sum for 3, and remember the sneaky composites in the 100–300 range that hide factors like 7, 11, 13, 17 and 19.

Which of the following is a prime number?

A171B161C193D177

The sum of three prime numbers is 90, and one exceeds another by 30. One of the numbers is:

A47B59C67D41

Which pair consists of twin primes?

A(15, 17)B(17, 19)C(21, 23)D(25, 27)