Divisibility Rules and Number Property Puzzles
Divisibility rules are quick checks that show whether one whole number divides exactly into another, without doing the full division. 11+ papers use them to solve puzzle-style questions that give several clues about a mystery number, for example that it is even, divisible by 3, and lies in a given range, where knowing the rules narrows down the possibilities fast instead of testing every number in turn.
Before you start
Make sure you're comfortable with these topics first:
Method
- Learn the rule for 2 (the last digit is even), for 5 (the last digit is 0 or 5), and for 10 (the last digit is 0).
- Learn the rule for 3 (the digits add up to a multiple of 3) and for 9 (the digits add up to a multiple of 9).
- Learn the rule for 4 (the last two digits form a number divisible by 4) and for 6 (the number is divisible by both 2 and 3).
- When a puzzle gives several clues about a mystery number, apply the rules one at a time to cross off numbers that fail a clue, rather than checking every number from scratch.
- Use factor and multiple language precisely: a factor divides exactly into a number with no remainder, while a multiple is a result of multiplying that number by a whole number.
- Once you have a candidate answer, check it against every clue in the question, not just the last one you used, before writing it down.
Worked example
I am a two-digit number. I am divisible by both 4 and 6. My digits add up to 6. I am greater than 50. What number am I?
- A number divisible by both 4 and 6 must be divisible by their lowest common multiple. Since 4 = 2 x 2 and 6 = 2 x 3, the lowest common multiple is 2 x 2 x 3 = 12.
- List the two-digit multiples of 12: 12, 24, 36, 48, 60, 72, 84, 96.
- Check which of these have digits adding up to 6: 24 (2 + 4 = 6) and 60 (6 + 0 = 6) both fit.
- Apply the final clue, greater than 50: only 60 fits, since 24 is less than 50.
- The answer is 60. Check: 60 / 4 = 15 exactly, 60 / 6 = 10 exactly, its digits add to 6, and it is greater than 50.
Practice questions
Type your answer and press Check to be marked straight away, or reveal the answer and mark yourself.
Q1Is 348 divisible by 4? Show your working.Show answer
Answer: Yes. The last two digits form 48, and 48 / 4 = 12 exactly, so 348 is divisible by 4.
Q2Is 522 divisible by 3?Show answer
Answer: Yes. Its digits add up to 5 + 2 + 2 = 9, and 9 is a multiple of 3.
Q3Which of these numbers is NOT divisible by 9: 342, 630, 451?Show answer
Answer: 451. Its digits add up to 4 + 5 + 1 = 10, which is not a multiple of 9 (342 and 630 both have digit sums of 9).
Q4Write down a multiple of both 4 and 5 that is less than 30.Show answer
Answer: 20 (the lowest common multiple of 4 and 5 is 20, and the next multiple, 40, is too big).
Q5A number is divisible by 8. Is it always divisible by 4? Explain.Show answer
Answer: Yes, because 8 = 4 x 2, so any number that divides exactly by 8 must also divide exactly by 4.
Q6Find the smallest three-digit number divisible by 6.Show answer
Answer: 102. It is even (divisible by 2) and its digits add to 3 (divisible by 3), so it is divisible by 6; 100 and 101 fail these checks.
Q7I am even, I am divisible by 3, and I lie between 20 and 30. What number am I?Show answer
Answer: 24 (being even and divisible by 3 means divisible by 6, and the only multiple of 6 strictly between 20 and 30 is 24).
Q8State the divisibility rule for 5.Show answer
Answer: A number is divisible by 5 if its last digit is 0 or 5.
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
N is a number between 500 and 600. N is divisible by both 8 and 9. The digits of N add up to less than 10. Find N, showing your method.
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Tick each line you got. Your score builds from the marks on the scheme.
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N is divisible by both 5 and 6, and lies between 300 and 400. The digits of N add up to 12. Find N, showing your method.
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Tick each line you got. Your score builds from the marks on the scheme.
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Free printable worksheet
Want more practice on paper? Download the divisibility rules and number property puzzles worksheet pack - 7 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 24 of 11+ Maths Workbook 4, the whole course as one free printable PDF.
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