Quandaries
and Queries 

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. 

Hi Lori, Base 5 is a nice example to use because you can start with pennies, nickles and quarters (25 is 55=5^{2}). 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
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
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
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 = 5^{3}.) Hence the base 5 number is
The final 4 divided by 5 is 0 with a remainder of 4 so, finally 613 written in base 5 is
I hope this helps, Penny 

