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Class 8 Number Play: cheat sheet
CBSE Mathematics. 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
| Division | a = bq + r, with 0 ≤ r < b |
| Divisibility by 11 | 8217: (8 + 1) − (2 + 7) = 0, so 11 divides it |
| Odd and even | odd + odd = even; odd × odd = odd |
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
Find the digit that goes in the box so that 72☐9 is divisible by 9.
Working: the other digits add up to 18; the total must be a multiple of 9
Answer: 0 or 9
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