Topics / Number Theory

Number Properties

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

Number PropertiesAMC 10 Step by Step

Name: Date: September 13, 2026

Number Theory · 32 problems have this as their main idea

primary-topic problems (2.5% of all)

What you need to know

Parity: odd $\pm$ odd $=$ even, odd $\times$ odd $=$ odd; a sum of integers is odd iff it has an odd number of odd terms. Squares satisfy $n^2 \equiv n \pmod 2$ .
Perfect squares end in $0, 1, 4, 5, 6, 9$ ; they are $\equiv 0, 1 \pmod 4$ and odd squares are $\equiv 1 \pmod 8$ . Squares $\equiv 0, 1 \pmod 3$ ; cubes $\equiv 0, \pm 1 \pmod 9$ .
Among any $k$ consecutive integers, exactly one is divisible by $k$ ; hence $k! \mid n(n+1)\cdots(n+k-1)$ .
Floor function: $\lfloor x \rfloor$ is the greatest integer $\le x$ , so $\lfloor x \rfloor \le x < \lfloor x \rfloor + 1$ . The number of perfect squares $\le N$ is $\lfloor \sqrt N \rfloor$ .
$n^2 - 1 = (n-1)(n+1)$ ; for $n$ odd this is divisible by $8$ , and for $3 \nmid n$ it is divisible by $3$ .
Sum of the first $n$ positive integers is $\frac{n(n+1)}{2}$ ; sum of the first $n$ odd numbers is $n^2$ .

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 residues (problems 10–16).
Counting squares or cubes in a range, or integers that are neither: floors of square and cube roots plus inclusion–exclusion.
Parity arguments: a sum of $2025$ odd numbers cannot be $2024$ ; a chessboard-style impossibility.
Sequences of integers with a growth rule; find the number of terms below $N$ or a specific term via floors.
"Which of the following must be true about $n$ ?" with five statements about odd/even, squares, consecutive products.

Standard approaches

Check parity first; it kills half the candidates in seconds.
Factor the expression, then examine each factor's divisibility separately ( $n(n+1)$ is always even, etc.).
Reduce mod $3, 4, 8, 9$ where squares and cubes have few residues.
For counts in $[1, N]$ , use floors and inclusion–exclusion, and double-check endpoints.
When a statement "must" hold, test $n = 1, 2$ , and a prime; a single 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) $50$

Since $n^2 - 1 = (n-1)(n+1)$ , divisibility by $24 = 8 \cdot 3$ can be checked separately. If $n$ is odd, $n - 1$ and $n + 1$ are consecutive even numbers, one of them a multiple of $4$ , so $8 \mid (n-1)(n+1)$ ; if $n$ is even, the product is odd. If $3 \nmid n$ , one of $n-1, n+1$ is a multiple of $3$ ; if $3 \mid n$ , neither is. So the condition is exactly $\gcd(n, 6) = 1$ . Among $1, \ldots, 100$ there are $50$ even numbers, $33$ multiples of $3$ , and $16$ multiples of $6$ , leaving $100 - 50 - 33 + 16 = 33$ . The answer is $\boxed{\textbf{(C)}\ 33}$ .

Pitfalls

Believing " $n^2 \equiv 1 \pmod 8$ " for all $n$ ; it holds only for odd $n$ .
Miscounting squares in $[a, b]$ : it is $\lfloor \sqrt b \rfloor - \lceil \sqrt a \rceil + 1$ , not $\lfloor \sqrt b \rfloor - \lfloor \sqrt a \rfloor$ when $a$ is a perfect square.
Applying $\lfloor x + y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor$ ; it can be off by one.
Forgetting that $n = 1$ or $n = 0$ are integers that may satisfy (or break) the condition.

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)

Problems, easiest first

2024 AMC 10A · #4Number Properties

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?

2023 AMC 10A · #3Number Properties

How many positive perfect squares less than $2023$ are divisible by $5$ ?

2016 AMC 10A · #4Number Properties

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 …

2016 AMC 10A · #5Number Properties

A rectangular box has integer side lengths in the ratio $1: 3: 4$ . Which of the following could be the volume of the box?

2013 AMC 10B · #4Number Properties

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 $n$ ?

2011 AMC 10A · #2Number Properties

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?

2009 AMC 10A · #2Number Properties

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?

2023 AMC 10B · #9Number Properties

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 $2023$ ?

2020 AMC 10B · #7Number Properties

How many positive even multiples of $3$ less than $2020$ are perfect squares?

2019 AMC 10A · #5Number Properties

What is the greatest number of consecutive integers whose sum is $45?$

2014 AMC 10A · #8Number Properties

Which of the following numbers is a perfect square?

2012 AMC 10B · #9Number Properties

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?

2006 AMC 10A · #10Number Properties

For how many real values of $x$ is $\sqrt{120-\sqrt{x}}$ an integer?

2006 AMC 10A · #9Number Properties

How many sets of two or more consecutive positive integers have a sum of $15$ ?

2004 AMC 10B · #6Number Properties

Which of the following numbers is a perfect square?

2003 AMC 10B · #7Number Properties

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 $\lfloor \sqrt{1} \rfloor + \lfloor \sqrt{2} \rfloor + \lfloor \sqrt{3} \rfloor + \cdots + \lfloor \sqrt{16} \rfloor.$

2025 AMC 10A · #21Number Properties

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 …

2023 AMC 10B · #15Number Properties

What is the least positive integer $m$ such that $m \cdot 2! \cdot 3!\cdot 4!\cdot 5! \dots 16!$ is a perfect square?

2022 AMC 10B · #14Number Properties

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?

2020 AMC 10A · #18Number Properties

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.)

2007 AMC 10A · #17Number Properties

Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$ . What is the minimum possible value of $m + n$ ?

2024 AMC 10A · #20Number Properties

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 $S$ ?

2023 AMC 10B · #22Number Properties

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 $x$ ?

2020 AMC 10A · #22Number Properties

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 $x$ .)

2020 AMC 10B · #24Number Properties

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 $x$ .)

2015 AMC 10B · #21Number Properties

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 …

2011 AMC 10A · #19Number Properties

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 …

2011 AMC 10B · #21Number Properties

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 $w$ ?

2002 AMC 10A · #22Number Properties

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?

2018 AMC 10B · #25Number Properties

Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$ . How many real numbers $x$ satisfy the equation $x^2 + 10,000\lfloor x \rfloor = 10,000x$ ?

2016 AMC 10B · #25Number Properties

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 $x \ge 0$ ?

2010 AMC 10A · #25Number Properties

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: …