11 plus Number Properties questions

What a Number Properties question looks like

Number Properties is the part of 11 plus maths that asks what sort of number you are dealing with: a multiple of six, a prime, a square, a cube, or something with a given number of factors. These are recall questions with a little checking attached, and they turn up inside longer problems all through a paper. Its nearest neighbour in 11+ Daily is Classification & Logic, which sets a checklist of rules and brings flat shapes in alongside numbers; here the subject is always a number, and the answer is sometimes a count rather than one of the numbers offered.

The easier questions test one idea directly: which of these is a multiple of six, which set of three numbers are all multiples of four, which is a square number, how many factors twelve has. The middle of the bank joins two ideas together, such as a number that is a multiple of both three and five, or one that is a factor of 48 and a multiple of six. One question draws two overlapping circles, multiples of four and multiples of six, and asks which number belongs where they cross.

The hardest questions stack three conditions or turn the idea around: a two-digit prime whose digits add up to eight, or a cube number under a hundred whose square root is not a whole number. In one variation three children each make a claim about numbers, and the question asks which of them are wrong, so a child has to find a number that breaks a claim.

Example one: counting factors

How many factors does 30 have? The options are 5, 4, 6, 7 and 8.

Work in pairs, starting at one and going up. One times 30. Two times 15. Three times 10. Four does not go into 30. Five times six. The next number to try is six, and six has already appeared, so the list is finished: 1, 2, 3, 5, 6, 10, 15, 30.

The answer is 8.

The tempting wrong option is 6. A child who finds 1 and 30, 2 and 15, 3 and 10 and then feels finished has six factors. Going up one number at a time until the pair meets in the middle forces you to test four and five.

Example two: two group sizes at once

Kofi can share all his marbles into bags of 10, or all of them into bags of 15, and each time every bag is full with none spare. What is the fewest marbles he could have? The options are 20, 150, 30, 45 and 60.

List the multiples of the bigger number, because that list is shorter: 15, 30, 45. Try each against ten. Fifteen does not divide by ten, but 30 divided by 10 is 3.

The answer is 30.

The wrong option that catches most children is 150, from multiplying ten by fifteen. That is a number both bag sizes fit into, but the question asks for the fewest, and 30 is smaller. Two more satisfy half the question: 20 for bags of ten only, 45 for bags of fifteen only. And 60 fits both but is not the smallest.

Example three: finding the prime

Which two-digit prime number has digits that sum to 8? The options are 26, 53, 35, 44 and 62.

Every option has digits adding to eight, so that rule removes nothing and the prime part does all the work. A prime has exactly two factors, itself and one, so any even number above two is out: 26, 44 and 62 go. That leaves 53 and 35. Numbers ending in five divide by five, and 35 is 5 times 7, so it is not prime. Check 53 properly: it is not even, its digits add to eight so three does not divide it, it does not end in nought or five, and seven does not go in either, since 7 times 7 is 49.

The answer is 53.

The trap is 35: odd, digits adding to eight, and easy to accept if times tables are shaky.

How to answer it

1. Count the conditions before you look at the options. Easier questions carry one, harder ones carry three, including quiet ones like “two-digit” and “less than 100”.
2. Say what the key word means in your own words. A factor divides in exactly. A multiple is what you land on counting in that number. A prime has exactly two factors. A square is a whole number times itself.
3. Start with the condition that clears the most options: odd or even usually does it, then ending in nought or five, then adding the digits to test for three.
4. Test what is left against every condition, including the ones you have already used. Wrong options here are built to pass one rule and fail another.
5. When the answer is a count, list in pairs and stop only when the pair meets in the middle.

Common mistakes

• Losing a factor pair when counting. Missing 6 and 10 turns the eight factors of 30 into six. Listing in order from one is the fix.
• Swapping factor and multiple. Asked for a factor of 48 that is also a multiple of six, children often pick 36 or 42. Both are multiples of six, and neither divides into 48.
• Assuming odd means prime. Nine, 15, 21, 27, 33 and 35 are all odd and none of them are prime. Each turns up as a wrong option somewhere.
• Thinking a multiple of two numbers is anything even. A multiple of both four and six has to be a multiple of 12, so 8 and 26 do not qualify. A multiple of both three and five has to be a multiple of 15.
• Confusing squares with cubes. Asked for a cube that is not a square, a child picks 64, which is both: 4 times 4 times 4, and also 8 times 8.

How to practise 11 plus Number Properties questions

Most of the marks lost here go on recall rather than reasoning, so the useful work is knowing the square numbers to 144, the cubes 8, 27, 64 and 125, and the primes below 30 well enough that they need no thinking about. After that it is the habit of checking all five options instead of the first one that fits, and ten focused minutes a day beats an hour at the weekend, which is the pattern we set out in how much 11 plus practice a child needs each day. When your child gets one wrong, ask them to say what the property means before anyone mentions the answer, because the error is nearly always in the definition rather than the arithmetic. 11+ Daily has a Number Properties project alongside the rest of the number work, on laptop, tablet or phone, and the full list sits on our 11 plus maths practice page. Every question is original to 11+ Daily, nothing is a past paper and nothing is licensed from another publisher, and the bank is re-checked in rounds.

Questions parents ask