KA-0051Complementary counting
3 points, difficulty 1 of 3Level 3–4about 60sHow many of the numbers from 1 to 20 are not multiples of 3?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Count the ones you do not want first.
2The strategy
The word not is a signal to count the opposite. Count how many multiples of 3 are in the range, then take them away from the size of the range.
3The full solution
The multiples of 3 up to 20 are 3, 6, 9, 12, 15, 18 - that is 6 of them. So 20 - 6 = 14 numbers are not multiples of 3.
Solution
The reliable way
Count what you do not want, then subtract. The multiples of 3 from 1 to 20 are 3, 6, 9, 12, 15 and 18 - six of them. Everything else is not a multiple of 3, so the answer is 20 - 6 = 14. The transferable idea: the word not almost always means count the complement.
The elegant way
Dividing 20 by 3 gives 6 remainder 2, so there are 6 multiples and 14 non-multiples without listing anything.
Why this is on the test: It puts the complementary-counting habit on an easy range, so the method is automatic before the harder at-least-one questions arrive.