Question 1
n = 396.
What is the largest integer divisor of 396 that is at most 44?
- 22
- 46
- 396
- 84
- 44
Show the worked solution
- 396 = 44 × 9, so 44 divides 396.
- 44 is the largest integer ≤ 44 that still divides 396.
- Answer: 44.
Exam path · Number Properties
Number properties questions, from remainder and range
Number properties questions ask for a remainder, how many even integers sit in a range, or the largest divisor of a number that stays under a cap. The remainder is what is left after division, never the quotient. An even count is half the range when both ends are even, and the largest divisor under a cap is not the number itself when the cap is smaller.
Prelims number-system questions in this family are short: a remainder, an even or odd count, the largest divisor under a stated cap, or how many primes lie between two numbers. They are one-step once the question is read as a division rather than as a hunt through the options.
Try it against the clock. Number Properties runs at 30s easy, 40s medium and 55s 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.
How to solve number properties questions from the remainder
-
Remainder is what is left, not how many times it divides
47 divided by 4 is 11 remainder 3. The quotient 11 is the printed near miss. The remainder is always smaller than the divisor.
-
Even counts from 1 through n
From 1 through n inclusive there are n/2 evens when n is even, and (n−1)/2 when n is odd. Counting every integer, odds included, reprints n.
-
Largest divisor at most k
Walk the divisors, or test downward from k. The number itself divides itself, but it is not allowed once the cap is smaller than the number. The cap is a legal answer only when it actually divides.
-
Counting the primes between two numbers
"Between" leaves out both end points. Skip the even numbers, then strike multiples of 3, 5, 7 and 11 — a number with no divisor up to its square root is prime. Odd composites such as 91 or 119 are the usual over-count.
Four short shapes
| What the paper asks | The move |
|---|---|
| When A is divided by B, the remainder | A − B × floor(A/B), and it is smaller than B |
| How many evens from 1 through n | n/2 if n is even, else (n−1)/2 |
| Largest divisor of n that is at most k | The greatest d ≤ k with n divisible by d |
| How many primes lie between two numbers | Skip evens, strike multiples of 3, 5, 7 |
3 real number properties questions, with worked solutions
Generated by the same engine that mints the ComputePrep 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 2
When 203 is divided by 12, what is the remainder?
What is the remainder?
- 12
- 16
- 1
- 8
- 11
Show the worked solution
- 203 = 12 × 16 + 11.
- Remainder: 11.
- Answer: 11.
Question 3
n = 144.
What is the largest integer divisor of 144 that is at most 36?
- 18
- 26
- 36
- 144
- 25
Show the worked solution
- 144 = 36 × 4, so 36 divides 144.
- 36 is the largest integer ≤ 36 that still divides 144.
- Answer: 36.
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.
Answering with the quotient
47 ÷ 4 is 11 remainder 3. Selecting 11 answers a different question. The remainder cannot be 4 or more.
Counting every integer as even
From 1 through 20 there are 20 integers and 10 evens. The full count is always on the board.
Picking n itself when the cap is smaller
180 divides 180, but a cap of 30 forbids it. The answer is the largest divisor that also respects the cap — 30, if 30 divides 180.
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.
When 47 is divided by 4, why isn't the remainder 11?
11 is the quotient: 4 goes into 47 eleven times. What is left is 47 − 44 = 3, and a remainder has to be smaller than 4.
How many even integers are there from 1 through 20 — why isn't the answer 20?
20 counts every integer in the range, odds included. The evens are 2, 4, …, 20: ten of them, which is 20 ÷ 2.
What is the largest divisor of 180 that is at most 30 — why isn't the answer 180?
180 divides 180, but the cap is 30, so 180 is illegal. 30 itself divides 180 (180 ÷ 30 = 6), and nothing larger than 30 is allowed, so 30 is the answer.
FAQ
What do number properties questions cover in bank prelims?
Remainders, even and odd counts in a range, the largest divisor under a cap, and how many primes lie between two numbers. They are the short number-system items, usually one mark.
How do you find a remainder quickly?
Divide and keep what is left. 47 ÷ 4 leaves 3, because 4 × 11 = 44 and 47 − 44 = 3. The remainder is always smaller than the divisor, which throws out several options before you divide.
How long should a number properties question take?
About 30 to 55 seconds. This drill enforces 30 seconds on easy, 40 on medium and 55 on hard, on the server.
Is this practice free?
Yes, free with no account. Every question is generated fresh, so the numbers are never the same twice.
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