Exam path · Integer Counting
Integer counting questions ask how many integers in a printed range are multiples of one number, how many positive factors a number has, or how many integers are divisible by one number but not another. Count with a closed form: floor of the top divided by the step, minus floor of the number just below the bottom. Do not list the integers.
Prelims number-system sets ask this as 'how many numbers from A to B are divisible by K', a factor count, or divisible by one number but not another. Listing the range is how the clock runs out. The count is one subtraction.
Try it against the clock. Integer Counting 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 many multiples of 3 from 6 through 28? floor(28/3) − floor(5/3) = 9 − 1 = 8. Both ends count when they are multiples.
Strictly between 10 and 40 means from 11 through 39. Use the same formula on that inner range. Forgetting to drop an end that is itself a multiple is the usual off-by-one.
Count the multiples of k, then subtract the multiples of the least common multiple of k and m. Subtracting the multiples of m on their own double-counts or under-counts when k and m share a factor.
If n = p^a × q^b, the number of positive factors is (a+1)(b+1). Odd factors ignore the factor 2. Square factors keep only the even exponents.
| What the paper asks | The move |
|---|---|
| Multiples of k from lo through hi | floor(hi/k) − floor((lo−1)/k) |
| Divisible by k but not by m | Multiples of k, minus multiples of lcm(k, m) |
| How many positive factors | (a+1)(b+1) from the prime powers |
| How many of those factors are odd, or squares | Drop the 2s, or keep even exponents only |
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.
How many 3-digit positive integers from 462 through 569, inclusive, are divisible by 18?
How many such integers are there?
How many 2-digit positive integers from 25 through 78, inclusive, are divisible by 19?
How many such integers are there?
How many 3-digit positive integers from 343 through 670, inclusive, are divisible by 47?
How many such integers are there?
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.
From 100 to 400 that is hundreds of integers. The closed form is two divisions. Listing is how a 40-second question becomes a skipped one.
From 6 through 28, 6 and 27 both count for multiples of 3. 'Through' and 'inclusive' mean both ends. 'Strictly between' means neither.
Divisible by 2 but not by 3 is not 'evens minus multiples of 3'. A multiple of 3 that is odd was never in the first count. Subtract multiples of 6.
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.
7 drops one end. floor(28/3) − floor(5/3) = 9 − 1 = 8. Both 6 and 27 are multiples and both sit inside an inclusive range.
33 counts every multiple of 3, including the odd ones, which were never in the 50 evens. Subtract multiples of 6 instead: 50 − 16 = 34.
6 is the count of prime factors with repeats, or a partial list. 36 = 2² × 3², so the factor count is (2+1)(2+1) = 9: 1, 2, 3, 4, 6, 9, 12, 18, 36.
For multiples of k from lo through hi, inclusive, the count is floor(hi ÷ k) − floor((lo − 1) ÷ k). Both ends count when they divide evenly. Strictly between drops both ends before you apply the same formula.
Yes, inside the number-system block: how many numbers between two limits are divisible by a given number, how many factors a number has, and divisible by one number but not another.
About 30 to 55 seconds. This drill enforces 30 seconds on easy, 40 on medium and 55 on hard, on the server.
Yes, free with no account. Every question is generated fresh, so the range is never the same twice.
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