Find all positive integers n such that 12n-119 and 75n-539 are both perfect squares. Let N be the sum of all possible values of n. Find N.

Let 75n - 539 = l^2 and 12n - 119 = k^2 . where n is a natural number.

Multiply 75n - 539 = l^2 by 4 to give 300n - 2156 =4l^2 and 12n - 119 = k^2 by 25 to give 300n - 2975 = 25k^2.

Subtract the two new expressions to give 4l^2 - 25k^2 = 819 which can be factorised (using the difference of two squares) to give (2l - 5k)(2l + 5k) = 819.

The prime factorisation of 819 is 3^2 * 7 * 13 There are five cases to consider. Dealing with the cases (noting that 2l - 5k < 2l +5k ) yields that n can only be 20 or 12.

Hence N = 20 + 12 = 32.

SK
Answered by Soham K. STEP tutor

7645 Views

See similar STEP University tutors

Related STEP University answers

All answers ▸

How would you prove the 'integration by parts' rule?


How can I integrate e^x sin(x)?


Let y=arcsin(x)/sqrt(1-x^2). Show that (1-x^2) y'-xy-1=0, and prove that, for all integers n>=0, (1-x^2)y^{n+2}-(2n+3)xy^{n+1} -(n+1)^2 y^{n}=0. (Superscripts denote repeated differentiation)


Show that i^i = e^(-pi/2).


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences