Prime Factorisation

Factor Trees and the Division Ladder

Uganda Primary 7 Mathematics
Topic 3 — Term I

Every Number Has a Prime Fingerprint

Every whole number has a unique prime factorisation:

All three share prime building blocks from — in different combinations.

Prime and Composite Numbers Defined

Prime number: has exactly two distinct factors — 1 and itself.

Examples:

Composite number: has more than two factors.

Example:

Special case: is neither prime nor composite.

The Primes up to Fifty

Is 2 the only even prime? Yes — every other even number has 2 as a factor.

Why the Factorisation Is Unique

Fundamental Theorem of Arithmetic (informal):

Every whole number greater than 1 can be written as a product of primes in exactly one way (ignoring order).

• — the only way
• It does not matter how you start the decomposition

Chunk 1 Practice — Prime or Composite?

Classify each number and list its factors:

(Hint for 51: does divide ? Check: , and . Yes!)

Factor Trees — The Method

Rules:

• Split the number into two factors (neither can be 1)
• Circle prime factors (leaves — they are done)
• Box composite factors (they need further splitting)
• Continue until every branch ends in a prime
• Multiply all circled primes: that is the factorisation

Factor Tree for 36 — First Split

Leaf primes:

Factor Tree for 36 — Second Split

Same leaf primes! Different tree, identical factorisation.

Factor Tree — Worked Example: 120

Verify: ✓

Your turn: draw the factor tree for 84 in your exercise book.

Writing the Result in Exponent Notation

After collecting leaf primes, write using powers:

Convention: exponent of 1 is omitted ().

Order: write primes in ascending order (smallest first).

The Division Ladder — Method

Alternative method — especially useful for larger numbers:

1. Write the number at the top of the ladder
2. Divide by the smallest prime that divides it
3. Repeat with each new quotient
4. Stop when the quotient is 1
5. The factorisation is the product of all left-column primes

Division Ladder Worked Example: 180

Divisors collected:

Division Ladder Worked Example: 504

Scaling the Ladder to Larger Numbers

Test primes in order:

Verify by Multiplying the Factors Back

Multiply the prime factors back to the original:

A mismatch means you skipped a prime or stopped early.

Your Turn — Factorise 360

Use the division ladder. Test primes in order, then verify.

• Divide by 2 as many times as it goes, then 3, then 5
• Finish by multiplying your primes back to 360

Misconception 1 — Stopping at a Composite

Error: (4 is not prime!)

Rule: check every leaf — split any that is not prime.

Two More Common Factorisation Errors

2 — 1 as a prime factor

3 — Skipping a prime in the ladder

For : reject , then try ( ✓), not :

Lesson Summary — Two Methods, One Fingerprint

Both give the same result — and both end with a verification multiply.