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Number Properties

Perfect squares/cubes, parity, consecutive integers, floor/ceiling, integer sequences

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Number PropertiesAMC 10 Step by Step

Name: Date: October 28, 2026

all 32 problems, easiest first

What you need to know

Parity (odd and even). Odd $\pm$ odd $=$ even, and odd $\times$ odd $=$ odd. A sum of integers is odd exactly when it has an odd number of odd terms. A number and its square are both odd or both even: $n^2 \equiv n \pmod 2$ . Example: $3+5+7+10$ has three odd terms, so it is odd ( $25$ ).
Perfect squares end in $0$ , $1$ , $4$ , $5$ , $6$ , or $9$ , never $2$ , $3$ , $7$ , or $8$ . Divided by $4$ , a square leaves remainder $0$ or $1$ ; divided by $3$ , also $0$ or $1$ . An odd square leaves remainder $1$ when divided by $8$ . Example: $2027$ ends in $7$ , so it is not a square; $9^2 = 81 = 8\cdot10+1$ .
Perfect cubes leave remainder $0$ , $1$ , or $8$ when divided by $9$ . Since $8 \equiv -1$ , we write this as $\equiv 0, \pm1 \pmod 9$ . Example: $4^3 = 64 = 9\cdot7+1$ and $5^3 = 125 = 9\cdot13+8$ .
Consecutive integers. Among any $k$ consecutive integers, exactly one is divisible by $k$ . Even more: the product of any $k$ consecutive integers is divisible by $k! = 1\cdot2\cdots k$ . Why: for positive integers, the product divided by $k!$ is a binomial coefficient, which counts something, so it is a whole number. Example: $5\cdot6\cdot7\cdot8 = 1680$ , and $1680 \div 4! = 70 = \binom84$ .
Floor function. $\lfloor x \rfloor$ (the floor of $x$ ) is the greatest integer less than or equal to $x$ : round down. So $\lfloor x \rfloor \le x < \lfloor x \rfloor+1$ . Example: $\lfloor 3.7 \rfloor = 3$ and $\lfloor -1.2 \rfloor = -2$ .
Counting squares. There are $\lfloor \sqrt N \rfloor$ perfect squares from $1$ to $N$ . Example: $\lfloor \sqrt{200} \rfloor = 14$ , since $14^2 = 196$ and $15^2 = 225$ .
$n^2-1 = (n-1)(n+1)$ . If $n$ is odd, this is divisible by $8$ . If $3$ does not divide $n$ , it is divisible by $3$ . Example: $11^2-1 = 120 = 8\cdot15 = 3\cdot40$ .
Two sums. $1+2+\dots+n = \frac{n(n+1)}{2}$ , and the first $n$ odd numbers add to $n^2$ . Example: $1+2+\dots+20 = 210$ and $1+3+5+7+9 = 25$ .

How AMC 10 tests it

"How many integers $n$ in a range make $n^2-1$ (or $n^3-n$ ) divisible by $24$ ?" Parity and remainders (problems 10–16).
Counting squares or cubes in a range, or numbers that are neither: floors of square and cube roots plus the Principle of Inclusion-Exclusion (PIE).
Parity arguments: a sum of $2025$ odd numbers can't be $2024$ ; chessboard-coloring arguments that something is impossible.
Sequences of integers with a growth rule: find how many terms are below $N$ , or a particular term, using floors.
"Which of the following must be true about $n$ ?" with five statements about odd and even, squares, and products of consecutive numbers.

Standard approaches

Check parity first; it rules out half the possibilities in seconds.
Factor the expression, then check what divides each factor. Example: $n(n+1)$ is always even, since one of two consecutive integers is even.
Work mod $3$ , $4$ , $8$ , or $9$ , where squares and cubes have few possible remainders.
To count from $1$ to $N$ , use floors and PIE, and double-check the endpoints.
When a statement "must" hold, test $n = 1$ , $n = 2$ , and a prime. One counterexample settles it.

Worked example

For how many integers $n$ with $1 \le n \le 100$ is $n^2 - 1$ divisible by $24$ ?

(A) $25$ (B) $30$ (C) $33$ (D) $34$ (E) $50$

Step 1: split $24$ . $n^2-1 = (n-1)(n+1)$ and $24 = 8\cdot3$ . Since $8$ and $3$ share no factor, check each one separately.

Step 2: the $8$ . If $n$ is odd, $n-1$ and $n+1$ are two even numbers in a row, like $10$ and $12$ . One is a multiple of $4$ and the other a multiple of $2$ , so their product is a multiple of $8$ . If $n$ is even, the product is odd.

Step 3: the $3$ . Exactly one of $n-1$ , $n$ , $n+1$ is a multiple of $3$ . If $3$ does not divide $n$ , it divides $n-1$ or $n+1$ . If $3$ divides $n$ , it divides neither.

Step 4: the condition. So $24$ divides $n^2-1$ exactly when $n$ is odd and not a multiple of $3$ , that is, when $\gcd(n, 6) = 1$ .

Step 5: count with PIE. From $1$ to $100$ there are $50$ even numbers, $33$ multiples of $3$ , and $16$ multiples of $6$ (counted twice). That leaves $100-50-33+16 = 33$ .

The answer is $\boxed{\textbf{(C)}\ 33}$ .

Pitfalls

Believing every square leaves remainder $1$ when divided by $8$ . Only odd squares do. Example: $6^2 = 36$ leaves $4$ .
Miscounting squares from $a$ to $b$ . The count is $\lfloor \sqrt b \rfloor - \lceil \sqrt a \rceil + 1$ , where $\lceil x \rceil$ (the ceiling) means round up. Using $\lfloor \sqrt b \rfloor - \lfloor \sqrt a \rfloor$ leaves out $a$ when $a$ is a perfect square. Example: from $16$ to $50$ there are $4$ squares ( $16, 25, 36, 49$ ): $7-4+1 = 4$ , not $7-4 = 3$ .
Assuming $\lfloor x+y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor$ ; it can be off by one. Example: $\lfloor 0.6+0.6 \rfloor = 1$ , but $\lfloor 0.6 \rfloor + \lfloor 0.6 \rfloor = 0$ .
Forgetting that $n = 0$ and $n = 1$ are integers that may satisfy (or break) the condition. Example: in the worked example $n = 1$ counts, since $1^2-1 = 0$ and $24$ divides $0$ .

Traps that recur

Counting one solution for every integer n with |x| < 100 (n = -100 to 99, giving 200) without noticing that n = 99 yields only f = 1, which is excluded. (2018 AMC 10B #25)
Counting the 45 pairs (j, k) with 1 <= j < k <= 10 as separate jump points, forgetting that equal fractions like 1/2 = 2/4 = 3/6 are one jump. (2016 AMC 10B #25)
Forgetting the constraint r <= 2m (so that m^2 is really the largest square below n) and choosing a smaller m, or miscounting: 8 numbers in the sequence means 7 subtractions. (2010 AMC 10A #25)
Using the naive pattern 1, 4, 7, 10, … (odds only 6 apart) or evens spaced 4 apart (506), or dropping the extra 2 elements from the leftover 2021–2024. (2024 AMC 10A #20)

1.2024 AMC 10A #4
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
1. A) $20$
2. B) $21$
3. C) $22$
4. D) $23$
5. E) $24$
2.2023 AMC 10A #3
How many positive perfect squares less than $2023$ are divisible by $5$ ?
1. A) $8$
2. B) $9$
3. C) $10$
4. D) $11$
5. E) $12$
3.2016 AMC 10A #4
The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$ where $\left \lfloor \dfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\dfrac{x}{y}$ . What is the value of $\text{rem}\left(\dfrac{3}{8}, -\dfrac{2}{5}\right)$ ?
1. A) $-\frac{3}{8}$
2. B) $-\frac{1}{40}$
3. C) $0$
4. D) $\frac{3}{8}$
5. E) $\frac{31}{40}$
4.2016 AMC 10A #5
A rectangular box has integer side lengths in the ratio $1: 3: 4$ . Which of the following could be the volume of the box?
1. A) $48$
2. B) $56$
3. C) $64$
4. D) $96$
5. E) $144$
5.2013 AMC 10B #4
When counting from $3$ to $201$ , $53$ is the $51^\mathrm{st}$ number counted. When counting backwards from $201$ to $3$ , $53$ is the $n^\mathrm{th}$ number counted. What is $n$ ?
1. A) $146$
2. B) $147$
3. C) $148$
4. D) $149$
5. E) $150$
6.2011 AMC 10A #2
A small bottle of shampoo can hold 35 milliliters of shampoo, whereas a large bottle can hold 500 milliliters of shampoo. Jasmine wants to buy the minimum number of small bottles necessary to completely fill a large bottle. How many bottles must she buy?
1. A) $11$
2. B) $12$
3. C) $13$
4. D) $14$
5. E) $15$
7.2009 AMC 10A #2
Four coins are picked out of a piggy bank that contains a collection of pennies, nickels, dimes and quarters. Which of the following could not be the total value of the four coins, in cents?
1. A) $15$
2. B) $25$
3. C) $35$
4. D) $45$
5. E) $55$
8.2023 AMC 10B #9
The numbers $16$ and $25$ are a pair of consecutive positive squares whose difference is $9$ . How many pairs of consecutive positive perfect squares have a difference of less than or equal to $2023$ ?
1. A) $674$
2. B) $1011$
3. C) $1010$
4. D) $2019$
5. E) $2017$
9.2020 AMC 10B #7
How many positive even multiples of $3$ less than $2020$ are perfect squares?
1. A) $7$
2. B) $8$
3. C) $9$
4. D) $10$
5. E) $12$
10.2019 AMC 10A #5
What is the greatest number of consecutive integers whose sum is $45?$
1. A) $9$
2. B) $25$
3. C) $45$
4. D) $90$
5. E) $120$
11.2014 AMC 10A #8
Which of the following numbers is a perfect square?
1. A) $\dfrac{14!15!}2$
2. B) $\dfrac{15!16!}2$
3. C) $\dfrac{16!17!}2$
4. D) $\dfrac{17!18!}2$
5. E) $\dfrac{18!19!}2$
12.2012 AMC 10B #9
Two integers have a sum of 26. When two more integers are added to the first two integers the sum is 41. Finally when two more integers are added to the sum of the previous four integers the sum is 57. What is the minimum number of odd integers among the 6 integers?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
13.2006 AMC 10A #10
For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?
1. A) $3$
2. B) $6$
3. C) $9$
4. D) $10$
5. E) $11$
14.2006 AMC 10A #9
How many sets of two or more consecutive positive integers have a sum of $15$ ?
1. A) $1$
2. B) $2$
3. C) $3$
4. D) $4$
5. E) $5$
15.2004 AMC 10B #6
Which of the following numbers is a perfect square?
1. A) $98! \cdot 99!$
2. B) $98! \cdot 100!$
3. C) $99! \cdot 100!$
4. D) $99! \cdot 101!$
5. E) $100! \cdot 101!$
16.2003 AMC 10B #7
The symbolism $\lfloor x \rfloor$ denotes the largest integer not exceeding $x$ . For example, $\lfloor 3 \rfloor = 3,$ and $\lfloor 9/2 \rfloor = 4$ . Compute $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor.$
1. A) $35$
2. B) $38$
3. C) $40$
4. D) $42$
5. E) $136$
17.2025 AMC 10A #21
A set of numbers is called sum-free if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x+y$ is not an element of the set. For example, $\{1,4,6\}$ and the empty set are sum-free, but $\{1,4,5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1,2,3,...,20\}$ ?
1. A) $8$
2. B) $9$
3. C) $10$
4. D) $11$
5. E) $12$
18.2023 AMC 10B #15
What is the least positive integer $m$ such that $m \cdot 2! \cdot 3!\cdot 4!\cdot 5! \dots 16!$ is a perfect square?
1. A) $30$
2. B) $30030$
3. C) $70$
4. D) $1430$
5. E) $1001$
19.2022 AMC 10B #14
Suppose that $S$ is a subset of $\left\{ 1, 2, 3, \ldots , 25 \right\}$ such that the sum of any two (not necessarily distinct) elements of $S$ is never an element of $S.$ What is the maximum number of elements $S$ may contain?
1. A) $12$
2. B) $13$
3. C) $14$
4. D) $15$
5. E) $16$
20.2020 AMC 10A #18
Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}.$ For how many such quadruples is it true that $a\cdot d-b\cdot c$ is odd? (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.)
1. A) $48$
2. B) $64$
3. C) $96$
4. D) $128$
5. E) $192$
21.2007 AMC 10A #17
Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$ . What is the minimum possible value of $m + n$ ?
1. A) $15$
2. B) $30$
3. C) $50$
4. D) $60$
5. E) $5700$
22.2024 AMC 10A #20
Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold:
- If $x$ and $y$ are distinct elements of $S$ , then $|x-y| > 2.$
- If $x$ and $y$ are distinct odd elements of $S$ , then $|x-y| > 6.$
What is the maximum possible number of elements in $S$ ?
1. A) $436$
2. B) $506$
3. C) $608$
4. D) $654$
5. E) $675$
23.2023 AMC 10B #22
How many distinct values of $x$ satisfy $\lfloor{x}\rfloor^2-3x+2=0,$ where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$ ?
1. A) $\text{an infinite number}$
2. B) $4$
3. C) $2$
4. D) $3$
5. E) $0$
24.2020 AMC 10A #22
For how many positive integers $n \le 1000$ is $\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$ not divisible by $3$ ? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$ .)
1. A) $22$
2. B) $23$
3. C) $24$
4. D) $25$
5. E) $26$
25.2020 AMC 10B #24
How many positive integers $n$ satisfy $\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$ (Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$ .)
1. A) $2$
2. B) $4$
3. C) $6$
4. D) $30$
5. E) $32$
26.2015 AMC 10B #21
Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$ ?
1. A) $9$
2. B) $11$
3. C) $12$
4. D) $13$
5. E) $15$
27.2011 AMC 10A #19
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. Which of the following is closest to the percent growth of the town's population during this twenty-year period?
1. A) $42\%$
2. B) $47\%$
3. C) $52\%$
4. D) $57\%$
5. E) $62\%$
28.2011 AMC 10B #21
Brian writes down four integers $w > x > y > z$ whose sum is $44$ . The pairwise positive differences of these numbers are $1, 3, 4, 5, 6,$ and $9$ . What is the sum of the possible values for $w$ ?
1. A) $16$
2. B) $31$
3. C) $48$
4. D) $62$
5. E) $93$
29.2002 AMC 10A #22
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one?
1. A) $10$
2. B) $11$
3. C) $18$
4. D) $19$
5. E) $20$
30.2018 AMC 10B #25
Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ ?
1. A) $197$
2. B) $198$
3. C) $199$
4. D) $200$
5. E) $201$
31.2016 AMC 10B #25
Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$ , where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$ . How many distinct values does $f(x)$ assume for $x \ge 0$ ?
1. A) $32$
2. B) $36$
3. C) $45$
4. D) $46$
5. E) $\text{infinitely many}$
32.2010 AMC 10A #25
Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$ , then his sequence contains $5$ numbers:
$\begin{array}{ccccc} {}&{}&{}&{}&55\\ 55&-&7^2&=&6\\ 6&-&2^2&=&2\\ 2&-&1^2&=&1\\ 1&-&1^2&=&0\\ \end{array}$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$ ?
1. A) $1$
2. B) $3$
3. C) $5$
4. D) $7$
5. E) $9$

Number Properties — answersAMC 10 Step by Step

Name: Date: October 28, 2026

Keep this page separate from the problems.

Answers

#	Answer	The idea it turns on
1	B	The biggest two-digit number is 99. Twenty of them make only 1980, which is short of 2024, but twenty 99s plus a 44 make exactly 2024.
2	A	A square is a multiple of 5 only when the number squared is a multiple of 5. Count 5, 10, 15, and so on while the square stays below 2023.
3	B	The division 3/8 divided by -2/5 is -15/16, a bit above -1. Rounding down gives -1, not 0, so the answer is 3/8 - 2/5 = -1/40.
4	D	Call the sides k, 3k and 4k. The volume is 12 times k cubed, so it must be 12 times a perfect cube. Only 96 = 12 times 8 works.
5	D	Counting down from 201 to 53 counts both end numbers, so the count is 201 minus 53, plus 1.
6	E	Fourteen small bottles hold only 490 milliliters, which is not quite 500, so she needs one more bottle.
7	A	Every choice is a multiple of 5, so the pennies must total a multiple of 5, which means none; four coins of at least 5 cents make at least 20.
8	B	Neighboring squares n squared and (n+1) squared always differ by 2n + 1. So count the values of n that make 2n + 1 at most 2023.
9	A	A square that is a multiple of 6 must be the square of a multiple of 6, so list 6 squared, 12 squared, and so on below 2020.
10	D	The integers may be negative. A run from -44 up to 44 adds to 0, so adding 45 on the end gives sum 45 with 90 numbers in the run.
11	D	The two factorials share a big common part, so each choice is a perfect square times (n+1)/2. Only 18/2 = 9 is a perfect square.
12	A	Split the totals into three pairs that add to 26, 15 and 16. Only the pair adding to 15, an odd number, is forced to contain an odd integer.
13	E	Call the integer k. Then the square root of x equals 120 − k^2, which cannot be negative, so k can be 0, 1, 2, up to 10: eleven values.
14	C	Sum = (how many numbers) times (their average). Try runs of 2, 3, 4, 5, 6 numbers and see which give 15 using only positive numbers.
15	C	Write the bigger factorial as the smaller one times the extra numbers. For example 100! is 100 times 99!, so 99! times 100! is (99!) squared times 100, a square.
16	B	The rounded-down square root stays the same between perfect squares: it is 1 three times, 2 five times, 3 seven times, then 4 once at 16.
17	C	Look at the largest number in the set. Pair up the smaller numbers so each pair adds to it; the set can hold at most one number from each pair.
18	C	Pair each factorial with the next, like 3! times 4! = (3!) squared times 4. What remains is a square times 8!, and 8! needs 2 times 5 times 7.
19	B	The numbers 13 to 25 work, since every sum is at least 26. Pairing smaller numbers that add to the largest number shows 14 is impossible.
20	C	The difference is odd when exactly one of the two products is odd. A product of two numbers from 0 to 3 is odd for only 4 of 16 pairs.
21	D	75 is 3 times 5 times 5. To make a perfect cube, m must add two more 3s and one more 5, so m = 45 and n = 15.
22	C	Ten numbers in a row hold at most 3 members; four would force two odd members 6 apart. The pattern 1, 4, 8 in each ten reaches 608.
23	B	Call the floor k. Then x = (k squared + 2)/3, and this x must really have floor k. Only k = 0, 1, 2, 3 pass that test.
24	A	The three floors are equal, so their sum is a multiple of 3, unless n divides 999 or 1000. Count those divisors, leaving out n = 1.
25	C	Call the common value k. Then n = 70k - 1000, and n must be at least k squared but less than (k+1) squared. Solve both conditions.
26	D	Jump counts are steps divided by 2 or 5, rounded up. Their difference stays close to 3/10 of the steps, so only 62 to 66 need checking.
27	E	Call 1991 a² and 2001 b² + 9. Then b² − a² = 141, so a = 70 or a = 22. Only 22 also makes 2011 a perfect square.
28	B	The biggest difference, 9, is w minus z. It is made of three gaps between neighbors, which must be 1, 3 and 5, with 1 in the middle.
29	C	Starting from a square number of tiles, two rounds always bring you to the next smaller square: 100, 90, 81, then 72, 64, and so on.
30	C	Split x into whole part n and decimal part. Each interval from n to n + 1 has one solution when (n + 1) squared is below 10,000.
31	A	Each term depends only on the part of x after the decimal point. As it grows, f steps up once at each fraction with bottom at most 10.
32	B	Work backwards. If m squared is the largest square below n, the leftover is at most 2m, so each smallest start is the previous one plus the smallest allowed square.

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.