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.
Soham K. A Level Maths tutor, 13 plus  Maths tutor, IB Maths tutor, G...

1 year ago

Answered by Soham, an Uni Admissions Test .STEP. tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£26 /hr

Ana S.

Degree: Physics (Bachelors) - Oxford, Balliol College University

Subjects offered:.STEP., Physics+ 3 more

-Oxbridge Preparation-

“A friendly and understanding tutor with three years experience in school clubs, summer camps and private tutoring.”

£36 /hr

Joe B.

Degree: Mathematics G100 (Bachelors) - Bath University

Subjects offered:.STEP., Maths+ 3 more

Further Mathematics
-Personal Statements-

“Hi! I'm Joe, a friendly, experienced and patient tutor with in-depth knowledge of both the old and new A Level Maths & Further Maths specifications.”

PremiumGeorge B. GCSE Maths tutor, A Level Maths tutor, A Level Further Math...
£28 /hr

George B.

Degree: Mathematics (Masters) - Warwick University

Subjects offered:.STEP., Maths+ 1 more

Further Mathematics

“Fourth year undergraduate at one of the top universities for Maths. Eager to tutor and help improve your grades.”

About the author

Soham K.

Currently unavailable: for regular students

Degree: Mathematics (Bachelors) - Cambridge University

Subjects offered:.STEP., Maths+ 1 more


“Mathematics is a beautiful subject and I always teach Mathematics as such. Mathematics we are introduced to in textbooks can come off as really dry and pointless - it is not so!  When I teach Mathematics, I wish to change the general ...”

MyTutor guarantee

You may also like...

Other Uni Admissions Test .STEP. questions

Show that if a polynomial with integer coefficients has a rational root, then the rational root must be an integer. Hence, show that x^n-5x+7=0 has no rational roots.

By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Show that substituting y = xv, where v is a function of x, in the differential equation "xy(dy/dx) + y^2 − 2x^2 = 0" (with x is not equal to 0) leads to the differential equation "xv(dv/dx) + 2v^2 − 2 = 0"

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

View Uni Admissions Test .STEP. tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss