Divisibility by 3 - Math Central

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Question from Cathleen, a student:

To math central. I have to do a maths extension question that I don't understand. At first I thought I did.
It is about the dividing by three. In one part of the question, it asks me to show that the rule of division by three does not work for 23142 with a little 5 down the bottom.
What doe base 5 mean? We first thought that the little 5 down the bottom meant multiplying y the power of five.
Can you please tell me what it means so I can finish this question?

Hi Cathleen,

Using our standard base 10 notation the number 5437 is five thousand, four hundred and thirty seven, or said using digits 5000 + 400 + 30 + 7. Using powers of 10 this is

5437 = 5 × 10³ + 4 × 10² + 3 × 10 + 7.

Numbers can be expressed in bases other than 10, that is using powers of the base. For example 3402₅ is the number

3402₅ = 3 × 5³ + 4 × 5² + 0 × 5 + 1 = 3 × 125 + 4 × 25 + 0 × 5 + 2 = 477.

I might write this 477₁₀ but usually when we express numbers using the base 10 notation we don't use the subscript of 10.

The point in your maths extension question is that the test for divisibility by 3 is not valid if you express the number in base 5 notation. For example the sum of the digits of 3402₅ is 3 + 4 + 0 + 2 = 14 which is not divisible by 3 and yet 3402₅ = 477 and 477 is divisible by 3.

The number in your pr0blem is 23142₅ and the sum of its digits is divisible by 3. Express 23142₅ is base 10 notation and test to see if it is divisible by 3.

I hope this helps,
Penny

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