Vocabulary: a number consists of several digits. Numbers which multiply together to make a number are factors. So the factors of a number include the number itself and one. When the number of factors of N is exactly two, they can only be 1 & N; then N is prime.
Factors
Call your number to be tested N. Where you read N below, think “the number”.
It is obvious when N is even; look at the last digit; if that is even then N has a factor of two.
It is obvious when N has a factor of ten; look at the last digit; if that is a zero then N has a factor of ten.
If N ends in a zero or five it is a multiple of five. Five is therefore a factor of N.
An even number may be a multiple of four. Try halving it and see if that too is even. Then four is a factor of N.
Add the digits of N: if this total (the digit total) is a multiple of three, three is a factor of N. If the digit total is a multiple of nine, nine is a factor of N. If your digit total is large, add the digits again and use this second digit total.
Examples, showing the obvious factors, but not all, not even all the prime ones:
14450 has factors 2 & 5 & 10
14250 has prime factors 2, 3, 5, 10, (digit total is 12, so it is divisible by 3),
14256 has factors of 2 & 3 & 9 (digit total is 18, second digit total is 9 so 9 is a factor)
… and 4 because half is 7128, and eight because half of that is 3564 and 16 because half of that is 1728 but not 32 because the next half is 891. So 14256 could be written as 14256 = 16 x 9 x (something). This something is 99 which gives us factors of 9 and 11. So
14256 = 16 x 9 x 9 x 11. We might write this really succinctly (tidily, shortly) as
14256 = 2⁴ x 3⁴ x 11 where by 2⁴ I mean 2x2x2x2 and 3⁴ means 3x3x3x3. So the prime factors of 14256 are just 2, 3 and 11. This special form is now called canonical prime representation, but terms do vary and change. I was taught this was Euler’s format, for example, which is no longer used.
All the factors of 28 are 1, 2, 4, 7,14, 28. They come in pairs, 1x28, 2x14, 4x7. The factors add to make 28, so 28 is called perfect. I say there is just one perfect number with one digit, one with two [28], one with three and one with four. Challenge! (looking it up on the internet is cheating).
Exercises:
1 Find the single digit factors of: a) 105 b) 12345 c) 1044 d) 1089
2 Find all the factors of: a) 36 b) 63 c) 91.
3 Find the prime factors of: a) 12 b) 56 c) 2100, d) 144.
4 Write these in Euler’s format for prime factors: a) 144 b) 2100 c) 496.
5 a) From the definition, what is the first prime number?
b) Pick two adjacent primes and multiply them together to give an answer, N. How many factors does your resulting N have?
c) If you square your N, how many factors does N2 have?
6 Find the single digit factors of: a) 28 b) 12345679 c) 10044 d) 108090
7 Find all the factors of: a) 360 b) 343 c) 496.
8 Find the prime factors of: a) 1024 b) 456 c) 30030, d) 46189.
9 Write these in canonical representation format for prime factors: a) 4096 b) 1764 c) 8028.
10 Add the factors of 6 and 28 and 496. What do you notice about these three totals? If this is a series, what is the (four digit) next one? This could once have been a GCSE investigation.
14450 = 2.5².17² 14250 = 2.3.5³.19² 14256 = 2⁴.3⁴.11
1. 105= 1.3.5.7, 12345 = 1.3.5.823 1044= 1.2^2.261 (so 1,2,4 only), 1089=3².11², so 1,3,9 only
2. fac(36)={1,2,3,4,6,9,12,18,36}; fac(63) = {1,3,7,9,21,63}; fac(91) = {1, 7, 13, 91}
3. PriFac(12)={2,3} PriFac(56) = {2,7} PriFac(2100) = {2,3, 5, 7} PriFac(144) = {2,3}
5. (a) 2 is the first prime with exactly two distinct factors, 1 and 2. (b) four: 1,p,q,pq ( c) {1,p,pp,q,qq,pq,ppq,pqq,ppqq} so nine factors.
6. single digit factors only: 28->1, 2,4,7; 123456789 ->1,3,9; 10044 -> 1,2,3,4,6; 108090-> 1,2,3,4,5,6,9
7. fac(360) = {1,2,3,4,5,6,8,9, 10,12,15,18,20,24,30,36, 40,45,60,72,90,120,180,360} 24 of them. fac(343) = {1,7,49,343}; fac(496) = {1,2,4,8,16,31,62,124,248,496}
8 PriFac(1024)={2} PriFac(456) = {2,57}; PriFac(30030) = {2,3,5,1001} ; Prifac(46189) = {11, 13, 17,19}
9. 4096 = 2^12 = 212 one way of writing this is {12,0,0,0,…}
1764 = 2^2.11^2 = 2².11² or perhaps {2,0,0,0,2,0,0,…}
8028 = 2².3².223 223 is prime.
10. PriSum(6) = 1=2=3=6 = 12 = 2x6. PriSum(28 = 1=2=4=7+14+28 = 56 PriSum (496)=2x496.6 = 2¹.3 = 2¹.(2²-1) 28=2².7 = 2².(2³-1) 496=2⁴.31=2⁴.(2⁵-1) 2⁶.(2⁷-1) = 8028
factors of 8028 are 1,2,4,8,16,32,64, 127, 254, 508, 1016, 2032, 4064, 8128. I say there is no fve digit one but there is ‘a six'.
