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Method 1: Quotient and remainder
7*5 = 35
35/12 = 2 remainder 11
Imagine you had 35 cookies and 12 friends.
Each friend gets 2 whole cookies (quotient), meaning they eat 2*12 = 24 cookies so far.
That leaves 35-24 = 11 cookies left over (remainder).
We're one cookie short of everyone able to get 3 whole cookies.
Method 2: Repeated subtraction
7*5 = 35
35-12 = 23
23-12 = 11
Recall that the process of long division involves repeated subtraction.
Notice there are two lines of subtracting by 12, to indicate the quotient of 2 mentioned earlier.
Method 3: Modular arithmetic
7*2 = 14 = 2 (mod 12)
7*4 = 2*(7*2) = 2*2 = 4 (mod 12)
7*5 = 7+7*4 = 7+4 = 11 (mod 12)
Ignore method 3 if your teacher has not covered this topic yet.
Even if your teacher introduced this notation style, it might still be a bit cryptic. So feel free to pick your favorite method.
Modular arithmetic is considered clock arithmetic because the numbers wrap around at some point. We only worry about the remainder.
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