How do you know what to divide by while using a division ladder?

Answer 1

By following these steps, you do the prime factorisation using the ladder method; the steps show how the decision of the next divisor is made:

1. At the top of the ladder start by setting the divisor to the smallest Prime number (2).
2. If the divisor does not divide into the dividend, set it to the next higher prime number and go back to step 2.
3. Write the divisor in and divide by it to get the quotient
4. If the quotient is not 1 (and not a prime number) then using this quotient as the next dividend go to step 2.
5. The list of divisors written in is the prime factorisation of the number.

A reminder:

• Divisor - the number by which you are dividing
• Dividend - the number into which you are dividing
• Quotient - the result of the division.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

If you are not sure of the prime numbers, then start with 2, and when that fails, then use the odd numbers starting with 3 (that is the numbers 3, 5, 7, 9, 11, 13, 15, 17, ... in order) - not all of them are prime, but only the prime ones will divide in at any stage of the ladder as long as you use them in increasing size (or you make a mistake).

example: Prime factorisation of 1638 using the steps:

1. set divisor to 2

2. 1638 divisible by 2

3. 1638 ÷ 2 = 819

4. dividend becomes 819

2. 819 not divisible by 2, divisor becomes 3

2. 819 divisible by 3

3. 819 ÷ 3 = 273

4. dividend becomes 273

2. 273 divisible by 3

3. 273 ÷ 3 = 91

4. dividend becomes 91

2. 91 not divisible by 3, divisor becomes 5

2. 91 not divisible by 5, divisor becomes 7

2. 91 divisible by 7

3. 91 ÷ 7 = 13

4. dividend becomes 13

13 is prime and so the need to go back to step 2 is unnecessary, but if the prime numbers are not known too well (see above), then the ladder could continue:

2. 13 not divisible by 7, divisor becomes 9

2. 13 not divisible by 9, divisor becomes 11

But if 9 is known not to be prime, then the above two steps would become a single step:

2. 13 not divisible by 7, divisor becomes 11

and the ladder would continue:

2. 13 not divisible by 11, divisor becomes 13

2. 13 divisible by 13

3. 13 ÷ 13 = 1

4. quotient is 1

leading the result:

5. Prime factorisation is: 2 x 3 x 3 x 7 x 13

When the dividend became 13, knowing the [smaller] prime numbers would help as it would be realised that 13 is prime and so there was no need to check 7 and 11.

Drawn in ladder format, it looks like:

2 | 1638

...-------

..3 | 819

.....------

..3 | 273

.....------

....7 | 91

.......-----

.........13

Prime factorisation: 2 x 3 x 3 x 7 x 13