Level 3: Divisibility

Number theory · level 3 of 8 · rough age 9-10 (a guide for parents; anyone can start anywhere)

In level 2 you found factors by building rectangles and trying one number after another. In this level you learn quick tests that tell you whether one number divides another without doing the division, and, just as important, why each test works. Then you break numbers into their prime building blocks and use those blocks to count every factor a number has, without listing them.

Lesson	You will be able to
Divisibility by 2, 5 and 10	Test any number for 2, 5 and 10 from its last digit, find the remainder, and explain why only the last digit matters
Divisibility by 4 and 8	Test for 4 with the last two digits and for 8 with the last three, explain why, and work out leap years
Divisibility by 3 and 9	Use the digit sum to test for 3 and 9 and to find remainders, and explain the trick with 10 = 9 + 1
Divisibility by 6, 11 and others	Combine tests for 6, 12, 15 and more, use the alternating digit sum for 11, and test for 7
Prime factorization	Break any number into primes with a factor tree or a ladder, and write it with powers, like 360 = 2³ × 3² × 5
How many factors?	Count all the factors of a number from its prime factorization, and explain why the rule works

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