Difference between revisions of "2023 AMC 10B Problems/Problem 9"

Revision as of 06:48, 16 November 2023

Problem

The numbers 16 and 25 are a pair of consecutive postive 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 m be the square root of the smaller of the two perfect squares. Then, $(m+1)^2 - m^2 = m^2+2m+1-m^2 = 2m+1 \le 2023$ . Thus, $m \le 1011$ . So there are $\boxed{\text{(B)}1011}$ numbers that satisfy the equation.

~andliu766

Minor corrections by ~milquetoast

Solution 2

\textbf{Solution 2}

Take note that the pairs are defined by their first digit, so we try to find the pair $x$ and $x+1$ . Using the difference of squares factorization, we get the inequality $(2x+1) \leq 2023$ . Following from here, we get $x=1011$ , so there are $\boxed{\text{(B) 1011}}$ numbers that satisfy the equation.

~darrenn.cp

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

Solution 3

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 $1012^2-1011^2$ . Since 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

Video Solution 1 by SpreadTheMathLove

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

@@ Line 11: / Line 11: @@
 Minor corrections by ~milquetoast
+==Solution 2==
+\textbf{Solution 2}
+Take note that the pairs are defined by their first digit, so we try to find the pair <math>x</math> and <math>x+1</math>. Using the difference of squares factorization, we get the inequality <math>(2x+1) \leq 2023</math>. Following from here, we get <math>x=1011</math>, so there are <math>\boxed{\text{(B) 1011}}</math> numbers that satisfy the equation.
+~darrenn.cp
 Note from ~milquetoast: Alternatively, you can let m be the square root of the larger number, but if you do that, keep in mind that <math>m=1</math> must be rejected, since <math>(m-1)</math> cannot be <math>0</math>.
-==Solution 2==
+==Solution 3==
 The smallest number that can be expressed as the difference of a pair of consecutive positive squares is <math>3</math>, which is <math>2^2-1^2</math>. The largest number that can be expressed as the difference of a pair of consecutive positive squares that is less than or equal to <math>2023</math> is <math>2023</math>, which is <math>1012^2-1011^2</math>. Since these numbers are in the form <math>(x+1)^2-x^2</math>, which is just <math>2x+1</math>. These numbers are just the odd numbers from 3 to 2023, so there are <math>[(2023-3)/2]+1=1011</math> such numbers. The answer is <math>\boxed{\text{(B)}1011}</math>.

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