6
Factoring natural numbers into primes
Textbook, pp. 19–20
GoalLearn to factor a natural number into prime factors and write the result with powers.
New words
prime factor · tub ko‘paytuvchifactorization · ko‘paytuvchilarga ajratishpower (exponent) · darajacolumn method · ustun usuli
Explanation
A composite number can be written as a product of primes only. We divide the number by the smallest prime, divide the quotient by a prime again, and continue until the quotient is 1. The primes we divided by are the factors: 60 = 2 · 2 · 3 · 5. Equal factors are conveniently written as powers: 60 = 2² · 3 · 5. For each natural number there is only one such factorization (up to the order of factors).
Worked examples
84 : 2 = 42, 42 : 2 = 21, 21 : 3 = 7, 7 : 7 = 1. So 84 = 2 · 2 · 3 · 7 = 2² · 3 · 7.
360 = 2 · 2 · 2 · 3 · 3 · 5 = 2³ · 3² · 5. Check: 8 · 9 · 5 = 360.
Class activity
“Number tree”: the number 72 is written on the board. Pupils split it into two factors, draw branches and continue until all ends are prime. Groups that drew different trees compare their results.
Practice
1
Factor 90 into primes.
2 · 3 · 3 · 5
2
Factor 150 into primes and write with powers.
2 · 3 · 5²
3
What is the product 2 · 2 · 3 · 5 · 7?
420
4
Why must all factors be prime in a factorization? Is 12 = 3 · 4 enough?
No, 4 is not prime. We split 4 again into 2 · 2: 12 = 2 · 2 · 3.