2023 AMC 10B Problems/Problem 9

Problem

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 $2023$ ?

$\text{(A)}\ 674 \qquad \text{(B)}\ 1011 \qquad \text{(C)}\ 1010 \qquad \text{(D)}\ 2019 \qquad \text{(E)}\ 2017$

Solution 1

Let $x$ be the square root of the smaller of the two perfect squares. Then, $(x+1)^2 - x^2 =x^2+2x+1-x^2 = 2x+1 \le 2023$ . Thus, $x \le 1011$ . So there are $\boxed{\text{(B)}1011}$ numbers that satisfy the equation.

~andliu766

A very similar solution offered by ~darrenn.cp and ~DarkPheonix has been combined with Solution 1.

Minor corrections by ~milquetoast

Note from ~milquetoast: Alternatively, you can let $x$ be the square root of the larger number, but if you do that, keep in mind that $x=1$ must be rejected, since $(x-1)$ cannot be $0$ .

Solution 2

The smallest number that can be expressed as the difference of a pair of consecutive positive squares is $3$ , which is $2^2-1^2$ . The largest number that can be expressed as the difference of a pair of consecutive positive squares that is less than or equal to $2023$ is $2023$ , which is $1012^2-1011^2$ . These numbers are in the form $(x+1)^2-x^2$ , which is just $2x+1$ . These numbers are just the odd numbers from 3 to 2023, so there are $[(2023-3)/2]+1=1011$ such numbers. The answer is $\boxed{\text{(B)}1011}$ .

~Aopsthedude

Solution 3

The difference of 2 consecutive positive integer squares is always an odd number (proved from above). Therefore, the total number of cases is $\left\lfloor \frac{2023}{2} \right\rfloor$ = $\boxed{B.1011}$

~Pinotation

Solution 4

Notice that the difference of two consecutive squares can be represented as $(x + 1) ^ 2 - x ^ 2$ . Using the Difference of Squares Formula, we simplify to get $(2x + 1)(1) = 2x + 1$ The number given in the problem was 2023, and thus we find that $x = \frac{2023 - 1}{2} \Rightarrow \boxed{\text{(B) 1011}}$ .

Video Solution by Math-X (First understand the problem!!!)

https://youtu.be/EuLkw8HFdk4?si=qr7LZahoIbDMBxvq&t=1848

~Math-X

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=qrswSKqdg-Y

Video Solution

https://youtu.be/DK--SMnDSr0

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution by Interstigation

https://youtu.be/gDnmvcOzxjg?si=cYB6uChy7Ue0UT4L

2023 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

Page

Toolbox

Search