Hi Peter,

I'm going to illustrate by converting 432₅ to base 8. I an going to do it in two steps. First convert 432₅ to base 10 and then convert this base 10 number to base 8.

Converting 432₅ to base 10 just requires remembering what base 5 means.

432₅ = 4 × 5² + 3 × 5 + 2 = 4 × 25 + 15 + 2 = 117.

To put 117 into base 8 imagine you have 117 CDs. Put them is stacks of 8 disks each. You get 14 stacks of 8 disks with 5 remaining (that is 117 ÷ 8 = 14 with a remainder of 5). Now take the 14 stacks and put them in groups of 8. You get 1 group of 8 stacks and 6 stacks remaining ((that is 14 ÷ 8 = 1 with a remainder of 6). Thus you have

1 group of 8 stacks (1 × 8 × 8 disks)
6 stacks (6 × 8 disks) and
5 individual disks.

Hence 117 = 1 × 8² + 6 × 8¹ + 5 = 165₈

Here, more compactly is what I did

117 ÷ 8 = 14, remainder of 5
14 ÷ 8 = 1, remainder 6
1 ÷ 8 = 0, remainder 1

Read the remainders from bottom to top, 165₈

Penny