Number Play

Divisibility tests for 3, 4, 8, 9, 11 and 12, digital roots, remainders, and patterns in sums of consecutive numbers.

Key ideas

• By 3 or 9: the sum of the digits is divisible by 3 or 9.
• By 4: the number made by the last two digits is divisible by 4. By 8: the last three digits.
• By 11: (sum of digits in odd places) − (sum in even places) is 0 or a multiple of 11.
• By 12: divisible by both 3 and 4.
• The digital root (keep adding the digits) equals the remainder on dividing by 9, except that a multiple of 9 has root 9.
• The sum of an odd number of consecutive numbers is that count times the middle number.

Remember

Watch out

• Using only the last digit to test for 4 or 8.
• Testing 12 with 2 and 6 (they share a factor, so 18 passes but is not divisible by 12).
• Forgetting that a missing digit can have two answers, such as 0 and 9.

Worked example