Test of Mathematics Solution Subjective 38 - When 30 divides a prime
Join Trial or Access Free ResourcesJoin Trial or Access Free Resources
Test of Mathematics Solution Subjective 38 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.
Also see: Cheenta I.S.I. & C.M.I. Entrance Course
Show that if a prime number p is divided by 30, the remainder is either prime or 1.
Suppose p = 30Q + R
(here p is the prime, Q and R are quotient and remainders respectively when p is divided by 30).
For all primes less than 30, Q = 0 and R=p. So that satisfies the claim of this problem.
If p > 30, suppose the remainder R is composite (not a prime). Since R 6 and N > 6 then MN (=R) > 36 > 30. But R < 30. So both M and N cannot exceed 6. Suppose M < 6. Then M must be divisible by 2, 3, or 5. We consider the case when M is divisible by 2 (other cases are analogous).
Suppose M = 2M'
R = MN = 2M'N
p = 30Q + 2M'N
But then the right hand side is divisible by 2. Hence the left hand side is also divisible by 2. But that is not possible as p is a prime larger than 30.
Hence R cannot be composite. This implies R is either a prime or 1.
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's M.Stat Entrance. They ranked within the first 50 in the entire country in these entrances. I.S.I. M.Stat Entrance
In 2026, the following Cheenta students have been successful for Indian Statistical Institute's B.Stat Entrance and Chennai Mathematical Institute's B.Sc. Math Entrance. They ranked within the first 200 in the entire country in these entrances. Most of these students attended the problem solving workshops regularly, which happen 5 days every week. CMI B.Sc. Math Entrance […]
In 2025, 8 students from Cheenta Academy cracked the prestigious Regional Math Olympiad. In this post, we will share some of their success stories and learning strategies. The Regional Mathematics Olympiad (RMO) and the Indian National Mathematics Olympiad (INMO) are two most important mathematics contests in India.These two contests are for the students who are […]
Cheenta Academy proudly celebrates the success of 27 current and former students who qualified for the Indian Olympiad Qualifier in Mathematics (IOQM) 2025, advancing to the next stage — RMO. This accomplishment highlights their perseverance and Cheenta’s ongoing mission to nurture mathematical excellence and research-oriented learning.