Question

Find the largest integer n such that both n+496 and n+224 are perfect squares

Answer 1

800
800+496=1296
800+224=1024
where 1296 is square of 36
and 1024 is square of 32

Answer 2

Answer:

The value of n is 4265

Step-by-step explanation:

Given: n+496 and n+224

To find: n

Solution:

n + 496 = x²

n + 224 = y²

x² - y² = 496-224 = 272

( x + y ) ( x - y ) = 272

272 = 2^4 * 17

∴ Possible pairs of products to generate 272 are

( 1, 272 ) with x, y = no solution

( 2, 136 ) with x, y = 69, 67

( 4, 68 ) with x, y = 36, 32

( 8, 34 ) with x, y = 21, 13

( 16, 17 ) with x, y = no solution

∵ we get the largest n from 69, 67

∴ Substituting the value in given expression

we get,

n + 496 = x² = 69² yields n = 4265

n + 224 = y² = 67² yields n = 4265

Hence, The value of n is 4265

#SPH3