7+6 = 13. Divide this over 5 to get 13/5 = 2 remainder 3. The quotient 2 is something we don't care about. We only worry about the remainder. Therefore 7+6 = 3 (mod 5)

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2+1 = 3 (mod 5) for similar reasoning as above. 3/5 = 0 remainder 3.

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20 = 9 (mod 11) since 20/11 = 1 remainder 9. Imagine you had 20 cookies and 11 friends. Each friend would get 1 whole cookie (quotient) and there could be 9 left over (remainder).

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35/11 = 3 remainder 2

Or you could use repeated subtraction like so to find the remainder

35-11 = 24

24-11 = 13

13-11 = 2

The last result (2) is smaller than 11, so we stop here and this is the remainder.

Therefore, 35 = 2 (mod 11)

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Here's another way to think of it. Consider you have $35 in your pocket. Let's say a store is selling trinkets for $11 each. We can pose these key questions:

What is the most number of trinkets you can buy?
If you buy that max amount, how much will you have left over?

The answer to the first question is 3 trinkets because 3*11 = 33 dollars is under the budget of $35. The amount left over is 35-33 = 2 dollars which is the remainder. It's not larger than 11, so we cannot buy any more trinkets at this point.

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3 years ago