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Maths: Number Properties

▶  Watch: how to find the number that fits every clue7 min

Four steps in about seven minutes — one real question, worked step by step, then the traps to watch out for.

In these questions, you're given clues about a number: what kind of number it is, and how big it is. Your job is to find the one answer that fits every clue.

Sometimes the question is a power instead: a number with a small number written up high. There are five answers to choose from.

GL builds these questions so that each wrong answer passes all the clues but one. The video above and the worked examples below work through the same questions, with the options in the same order.

The method

Four steps: list every clue, turn each into a test, cross off at the first miss, check it passes them all.

  1. Step 1 of 4

    List every clue

    Pick out every fact the question gives you about the number. Each clue is a test the number must pass.
  2. Step 2 of 4

    Turn each into a test

    Make each clue a quick question you can ask of any number. Prime: only 1 and itself. Square: a number times itself.
  3. Step 3 of 4

    Cross off at the first miss

    Run each test on the answers, one clue at a time. As soon as an answer fails a test, cross it off: one failed clue is enough.
  4. Step 4 of 4

    Check it passes them all

    Make sure your answer passes every clue. The wrong answers pass every clue but one.

Worked example

James is thinking of a prime number between 30 and 60. Its digits add up to 8. What is James's number?

A. 17 B. 44 C. 47 D. 53 E. 71

  1. List every clue. Here there are three. It's between 30 and 60. It's prime. And its figures add up to 8. The answer has to pass all three.
  2. Turn each into a test. Between 30 and 60: is it bigger than 30, and smaller than 60? Prime: does it have only two factors, 1 and itself? For a number with two figures, try dividing by 2, 3, 5 and 7. If none of them goes in exactly, it's prime. And for the last clue, add the two figures together.
  3. Cross off at the first miss. Run each test on the answers, one clue at a time. Is it between 30 and 60? 17 is too small, and 71 is too big, so A and E are out. Is it prime? 44 is even, so 2 goes into it: 44 = 4 × 11, so B is out. 47 and 53 pass: 2, 3, 5 and 7 don't go into either of them. Do the figures add up to 8? 4 and 7 make 11, not 8, so 47 is out: C is out. 5 and 3 make 8.
  4. Check it passes them all. 53 is between 30 and 60, it's prime, and its figures make 8. Three clues, three passes. Now look back at the crossed ones. 17 and 71 are both prime, and their figures make 8. 44's figures make 8 too, and 47 is prime and in range. Every wrong answer passes all the clues but one. So the answer is D.

Answer: D — 53 (between 30 and 60, prime, and 5 + 3 = 8)

A second worked example

The same four steps work on a power.

What is 5²?

A. 7 B. 10 C. 20 D. 25 E. 125

  1. List every clue. Here there's just one: the small 2, written up high.
  2. Turn each into a test. The small 2 says how many 5s to multiply together. Two 5s: 5 × 5.
  3. Cross off at the first miss. 7 is 5 + 2, and 10 is 5 × 2, so A and B are out. 125 is 5 × 5 × 5. That's three 5s, so E is out. And 20 is five 4s, not five 5s, so C is out too.
  4. Check it passes them all. 5 × 5 = 25. That's two 5s, just as the small 2 says. So this one is D.

Answer: D — 25 (5² = 5 × 5)

Things to watch out for

  1. Check every clue, for every option. The wrong answers fail just one. In the first worked example, 17 and 71 are both prime numbers, and their figures add up to 8. Stop at the first clue that fits, and either of them looks right.
  2. "All" means all. Which statement is true of all three of these numbers: 9, 12 and 15? Take factors of 36. That's true of 9 and 12. But 36 ÷ 15 leaves 6, so 15 isn't a factor of 36. One number fails, so the whole statement fails. Multiples of 3 is true of every one.
  3. Odd doesn't mean prime. Which of these is a prime number: 65, 83, 77, 80 or 69? 65, 77 and 69 are all odd, but none of them is prime: 65 = 5 × 13, 77 = 7 × 11 and 69 = 3 × 23. 80 is even, so 2 goes into it. 2 is the only even prime. That leaves 83.
  4. Watch for NOT. Which number is NOT a factor of 108? 9, 12, 18 and 54 all go into 108: 12 × 9, 9 × 12, 6 × 18 and 2 × 54. Those four are true, so if you miss the NOT, you'll pick one of them. But 108 ÷ 24 leaves 12, so 24 is the one that isn't a factor.

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