My name is Lori and I'm trying to help my 6th grader with converting numbers like 82 to base 5 and 182 to base 12. We saw your examples on 613, but still are confused. 

Thanks for the help.
Lori

Hi Lori,

Base 5 is a nice example to use because you can start with pennies, nickles and quarters (25 is 55=5²). I am going to write 114 in base 5.

Imagine that you have 114 pennies. Arrange the pennies in piles of five. There are 22 piles of five pennies each, and then 4 more pennies. I know this because 114 divided by 5 is 22 with a remainder of 4. Replace each pile of five pennies by a nickle.

Arrange the 22 nickles in piles of five. 22 divided by 5 is 4 with a remainder of 2 so there are 4 piles of five nickles each and 2 more nickles. Replace each pile of five pennies by a quarter.

Thus you have

4(quarters) + 2(nickles) + 4(pennies)
that is
4 25 + 2 5 + 4
= 4 5² + 2 5 + 4

Thus 114, written in base 5 is 424. Notice that each time I divided by 5 it was the remainder that became a base 5 digit.

So lets try 613 using this method. 613 divided by 5 is 122 with a remainder of 3, thus 112 nickles and 3 pennies. Thus the base 5 number is

???3

112 divided by 5 is 24 with a remainder of 2, thus I have 24 quarters, 2 nickles and 3 pennies. Hence the base 5 number is

???23

24 divided by 5 is 4 with a remainder of 4. (The money analogy breaks down here because wse don't have a coin worth 5 quarters, that is 125 = 5³.) Hence the base 5 number is

???423

The final 4 divided by 5 is 0 with a remainder of 4 so, finally 613 written in base 5 is

4423

I hope this helps,

Penny

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