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Number properties

Number properties questions — count the terms, do not list them

Number properties questions ask you to count integers with a stated property or identify divisors within a bound. Counting is a formula, not a listing exercise: the number of multiples of d from 1 to n is n divided by d, rounded down. For divisor questions, pair each factor with its partner and work inward from the bound.

This is a category where the intended method takes five seconds and the instinctive method takes forty. Asked how many even integers lie between 1 and 31, most candidates begin counting. The formula answers it immediately, and the same counting logic covers multiples of any number, which is most of what this topic asks.

Try it against the clock. Number Properties runs at 90s easy, 105s medium and 120s hard. The clock is stamped and judged on our server, so the limit you see is the deadline that is actually enforced — and it starts when you tap Start, not while the question is loading.

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How to answer number properties questions by counting, not listing

  1. Count multiples with a single division

    The evens from 1 to 31 are the multiples of 2, and there are 31 divided by 2, rounded down, which is 15. No listing and no sequence formula required.

  2. Shift the range before counting from a non-zero start

    For multiples of d from a to b, count them up to b and subtract the count up to a minus 1. Subtracting the count up to a itself is the classic off-by-one.

  3. Find large divisors by pairing from the small end

    To get the largest divisor of 60 at or below 30, note that 60 equals 30 times 2, so 30 divides 60 and nothing between 30 and 60 can. Every divisor above the square root pairs with one below it.

  4. Use the endpoints as a sanity check

    Inclusive ranges are where these questions are won or lost. Test your formula on a tiny case — evens from 1 to 4 should give 2 — before trusting it on the real numbers.

Counting integers with a property in a range

The question asksCompute
Evens from 1 to nn ÷ 2, rounded down
Multiples of d from 1 to nn ÷ d, rounded down
Multiples of d from a to b⌊b ÷ d⌋ − ⌊(a−1) ÷ d⌋
Odds from 1 to nn minus the count of evens
Largest divisor of n at most kPair from the small side of √n

3 sample number properties questions, with worked solutions

Generated by the same engine that mints the Quant Daily daily. Each answer is computed from the numbers printed in the question, and the walkthrough below each one is the engine's own working — not a solution written afterwards. Reload this page's live drill and you get different numbers.

Question 1

n = 216.

What is the largest integer divisor of 216 that is at most 36?

  1. 216
  2. 38
  3. 18
  4. 21
  5. 36
Show the worked solution
  1. 216 = 36 × 6, so 36 divides 216.
  2. 36 is the largest integer ≤ 36 that still divides 216.
  3. Answer: 36.

Question 2

When 62 is divided by 3, what is the remainder?

What is the remainder?

  1. 1
  2. 3
  3. 20
  4. 4
  5. 2
Show the worked solution
  1. 62 = 3 × 20 + 2.
  2. Remainder: 2.
  3. Answer: 2.

Question 3

How many even integers are there from 1 through 33 inclusive?

How many even integers are there?

  1. 17
  2. 56
  3. 33
  4. 29
  5. 16
Show the worked solution
  1. The evens are 2, 4, …, 32.
  2. That is 16 integers.
  3. Answer: 16.

Common mistakes this drill is built from

The wrong options are not random numbers. Each one is the result of a specific careless error for this topic, so picking one tells you which habit is costing you marks.

Listing the terms out

Writing 2, 4, 6 and so on is accurate and slow, and it is where transcription slips enter. The division does the same job in one step.

Subtracting the wrong endpoint

For a range starting at a, subtract the count up to a minus 1, not up to a. Subtracting the count up to a drops one valid multiple whenever a is itself a multiple.

Searching downward for a large divisor

Testing 30, 29, 28 in turn is slow. Divisors come in pairs around the square root, so checking the small partner finds the large one immediately.

Why your answer came out wrong

The specific wrong numbers this topic produces, and what each one tells you about the step you took. If you have just got a question wrong and want to know which habit did it, start here.

Are there 16 even integers from 1 to 31?

There are 15. The count is 31 divided by 2 rounded down, which is 15 — the last even number in range is 30. Sixteen comes from rounding up or from mistakenly including 32, and it is a printed option in questions built on this range.

For multiples of 3 from 6 to 30, is the answer ⌊30÷3⌋ − ⌊6÷3⌋ = 8?

That gives 8 and drops the 6 itself, which is a valid multiple. Subtract the count up to 5, not up to 6: ⌊30÷3⌋ − ⌊5÷3⌋ = 10 − 1 = 9. Using one below the lower bound is what keeps the range inclusive.

Is 29 the largest divisor of 60 that is at most 30?

29 is prime and does not divide 60 at all. The answer is 30, because 60 equals 30 times 2. Scanning downward from the limit invites exactly this kind of guess — pair from the small side instead and the answer appears on the first try.

FAQ

How many even integers are there from 1 to 31?

Fifteen. The evens are the multiples of 2, and the count of multiples of 2 up to 31 is 31 divided by 2 rounded down, which is 15. The same division works for multiples of any number, so it is worth using as the default rather than counting.

How do I count multiples in a range that does not start at 1?

Count the multiples up to the upper bound, then subtract the count up to one below the lower bound. Using one below the lower bound is what makes the range inclusive of its own starting point, and it is the step that removes the off-by-one.

What is the fastest way to find a large divisor under a limit?

Work from the small side. Every divisor above the square root has a partner below it, so scanning small factors and taking their partners finds the large divisors quickly. For 60 under 30, the partner of 2 is 30, and that is your answer straight away.

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