Every number's fingerprint
Tonight's idea
A factor of a number divides into it exactly. 6 is a factor of 60, because 60 = 6 × 10.
A prime is a number bigger than 1 that cannot be split into two smaller factors. The first few are 2, 3, 5, 7, 11, 13.
A factor tree breaks a number down until only primes are left. Split 60 into 6 × 10. Split 6 into 2 × 3 and 10 into 2 × 5. Circle each prime, because that branch has ended.
Now start a different way, with 60 = 4 × 15. The tree looks different. The circled primes are the same: two 2s, one 3, one 5.
This always happens. Every whole number bigger than 1 has one list of primes that multiply to make it. No other number has the same list. So the list works like a fingerprint.
Watch one: the fingerprint of 84
84 is even, so split off a 2: 84 = 2 × 42. Circle the 2.
42 = 6 × 7. Circle the 7. Then 6 = 2 × 3. Circle both.
The circled primes are 2, 2, 3 and 7, so
\[84 = 2 \times 2 \times 3 \times 7.\]Check by multiplying back: 2 × 2 = 4, 4 × 3 = 12, 12 × 7 = 84.
Find every prime up to 100 by crossing out the numbers that are not: mathdetours.com/demos/sieve.html.
Your turn
- Draw a factor tree for each number. Then write its fingerprint. (a) 12 (b) 18 (c) 30
- Draw two trees for 36. Start one with 6 × 6 and the other with 4 × 9. Do they end with the same primes?
- A number has the fingerprint 2 × 3 × 3 × 5. (a) What is the number? (b) Is 15 a factor of it? (c) Is 4 a factor of it? Use the fingerprint to explain.
- The fingerprint of 100 is 2 × 2 × 5 × 5. Do not draw a new tree. Write the fingerprint of (a) 200, (b) 300, (c) 1000.
Stretch: the longest fingerprint
32 = 2 × 2 × 2 × 2 × 2 has five primes in its fingerprint. Two numbers below 100 have six. Find them both. Why can no number below 100 have seven?
Answers
- (a) \(12 = 2 \times 2 \times 3\). (b) \(18 = 2 \times 3 \times 3\). (c) \(30 = 2 \times 3 \times 5\). Your trees may look different from someone else's. The primes will match.
- 6 × 6 gives 2 × 3 and 2 × 3. 4 × 9 gives 2 × 2 and 3 × 3. Both end with two 2s and two 3s: \(36 = 2 \times 2 \times 3 \times 3\).
- (a) 90. (b) Yes. 15 = 3 × 5, and a 3 and a 5 are both in the fingerprint. The rest, 2 × 3, is 6, and 15 × 6 = 90. (c) No. 4 = 2 × 2 needs two 2s and the fingerprint has only one.
- (a) 200 = 2 × 100, so add a 2: \(2 \times 2 \times 2 \times 5 \times 5\). (b) 300 = 3 × 100: \(2 \times 2 \times 3 \times 5 \times 5\). (c) 1000 = 10 × 100, and 10 = 2 × 5: \(2 \times 2 \times 2 \times 5 \times 5 \times 5\).
Stretch. 64 = 2 × 2 × 2 × 2 × 2 × 2 and 96 = 2 × 2 × 2 × 2 × 2 × 3. The smallest prime is 2, so the smallest number with seven primes is seven 2s multiplied together. That is 128, which is too big.