Divisibility rules

A divisibility rule is a quick test that tells you whether a number divides evenly by a given divisor, using only the number's digits. Instead of dividing 4,712 by 8 by hand, you can apply a rule and know the answer in seconds — a major time-saver on timed exams and in competition math.

The most useful rules: a number is divisible by 2 if its last digit is even; by 3 if its digit sum is divisible by 3; by 4 if its last two digits form a number divisible by 4; by 5 if it ends in 0 or 5; by 6 if it passes both the 2 and 3 tests; by 8 if its last three digits form a number divisible by 8; by 9 if its digit sum is divisible by 9; and by 10 if it ends in 0. For 7 and most larger primes, there is no simple digit test — dividing directly is usually fastest.

Take 4,716 as an example. Its digit sum is 4 + 7 + 1 + 6 = 18, which is divisible by 9, so 4,716 is divisible by 9 (and by 3). Its last two digits, 16, are divisible by 4, so the whole number is divisible by 4. Rules like these speed up prime factorization, reducing fractions, finding greatest common factors, and spotting whether a large number could be prime.

Divisibility rules appear throughout the AMC competition exams, where number theory problems often hinge on recognizing factors quickly, and on the Praxis Core Math exam under properties of whole numbers. Memorizing the rules for 2 through 10 pays off immediately on both.