Powers, Indices, LCM and GCF
Primary 6 · Mathematics
Powers, Indices, LCM and GCF
Lesson 3 of 6
30 min
Let's begin
Every whole number bigger than 1 is built from prime numbers, the way a house is built from bricks. Today you will break numbers down into their prime “bricks” and write them neatly using powers.
A prime number has exactly two factors: 1 and itself. The primes up to 30 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Write the number at the top. Split it into any two factors. Circle a factor when it is prime. Keep splitting the others until every branch ends in a circled prime.
Multiply the circled primes: 24 = 2 × 2 × 2 × 3. In index form: 24 = 2³ × 3.
Divide by the smallest prime that goes in exactly. Keep dividing by that prime until it no longer works, then move to the next prime. Stop when you reach 1.
Worked example
360
| Divide by | Number |
|---|---|
| 2 | 360 |
| 2 | 180 |
| 2 | 90 |
| 3 | 45 |
| 3 | 15 |
| 5 | 5 |
| 1 |
The primes on the left give 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
Worked example 1
If 2ⁿ = 32, find n. List the powers of 2: 2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32. So n = 5.
Worked example 2
If 3² × 3ⁿ = 3⁶, find n. Multiplying means adding indices: 2 + n = 6, so n = 4.
Worked example 3
If 7ⁿ ÷ 7² = 7³, find n. Dividing means subtracting indices: n − 2 = 3, so n = 5.
Worked example 4
If 72 = 2ᵃ × 3ᵇ, find a and b. The ladder gives 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3², so a = 3 and b = 2.
Tip
Start small. Always try dividing by 2 first, then 3, then 5, then 7. When a prime stops working, move on and never go back.
Work each one in your exercise book first, then tap Show answer to check.
Is 51 a prime number?
Answer: No. 51 = 3 × 17, so it has more than two factors.
Write 100 as a product of primes in index form.
Answer: 100 = 2 × 2 × 5 × 5 = 2² × 5².
Write 84 as a product of primes in index form.
Answer: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.
If 4ⁿ = 64, find n.
Answer: n = 3, because 4 × 4 × 4 = 64.
If 5ⁿ ÷ 5⁴ = 5³, find n.
Answer: n − 4 = 3, so n = 7.
What you learned today
Sources and credits
The stories, Rwandan worked examples and practice questions were written by Ganzaa.
Quick check
Which of these is a prime number?
