Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
7.4k views
in Olympiad by (65.8k points)

Let p, q be prime numbers such that n3pq – n is a multiple of 3pq for all positive integers n. Find the least possible value of p + q.

1 Answer

+1 vote
by (71.2k points)
selected by
 
Best answer

n3pq – n = 0 (mod 3)

n3pq – n = 0 (mod p)

n3pq – n = 0 (mod q)

∴ We need to fulfill following conditions :

(i) (3 – 1) | (pq – 1) ⇒ pq is odd

(ii) (p – 1) | (3q – 1)

Now 3 must not divide (p – 1) as it doesn’t divide (3q – 1)

∴ p – 1 = 3k + 1 or 3k + 2, for some integer k.

∴ ⇒ p = 3k + 2 or 3k + 3

But p ≠ 3k + 3 (as its prime)

∴ p = 3k + 2 clearly p > 3 and k = odd = 2λ +

1 (say)

⇒ p = 6λ + 5

(ii) (q – 1) | (3p – 1) so q will also be 5 (mod 6)

by trial least values for p & q are 17 and 11.

∴ p + q = 28

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...