Systematic Prime Factorisation

Using Number Properties

Jules tries to factorise 16 709. He uses divisibility rules and sees 2, 3, 5 are not factors of 16 709. Why does he not test whether 4 and 6 are factors?
(Answer: If 2 is not a factor then 4 cannot be a factor. If 2 is not a factor then 6 cannot be a factor.)

Why does Jules test for 7 next?
(Answer: 7 is the next prime number after 5.)

Jules works out 16 709 ÷ 7 = 2387 on a calculator. He searches for a prime factor of 2387. Does he need to check 2, 3, 5 or 7 again?
(Answer: He does not need to check 2, 3 or 5 as they are not factors of the original number. But 7 is a factor of 16 709, and so it might also be a factor of 2387.)

Explain how Jules knows 16 709 factorises in primes to 7 x 7 x 11 x 17.
(Answer: 2387 ÷ 7 = 341 so 7 is again a factor. Checking 7 again 341 ÷ 7 is not a whole number. The next prime is 11, and 341 ÷ 11 = 31. As 31 is prime the procedure stops. So 16 709 = 7 x 7 x 11 x 31.)

Examples: Find the prime factorisations of these numbers: 646       19 530
527    353       5439              6273        2136        127 400          559 000

Example: Kevin buys packets of biscuits for a school camp. He counts them and finds he has 899.Realistically there are two possible numbers of biscuits in a packet. What are the numbers?

Understanding Number Properties:

If g is a factor of a explain why a ÷ g is also a factor of a. Use this to explain this true statement: If a has no factors that are less than or equal to √ a, then a is a prime number.