There is an element of order 51 in the multiplicative group (Z/103Z)
True
Discussion: First note that (Z/103Z) has 102 elements as 103 is a prime (in fact one of the twin primes of 101, 103 pair). Also 102 = 2317. So it has Sylow-3 subgroup of order 3 (prime order hence it is cyclic too) and a Sylow-17 subgroup (which is similarly cyclic). Since (Z/103Z) is abelian all it's subgroups are normal. Thus product of Sylow-3 and Sylow-17 subgroups is a subgroup (direct product of normal subgroups is a subgroup) containing 51 elements which is again cyclic. Hence there is an element of order 51 (generator of this subgroup).
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.
Cheenta students shine at the Purple Comet Math Meet 2025 organized by Titu Andreescu and Jonathan Kanewith top national and global ranks.
Celebrate the success of Cheenta students in the Stanford Math Tournament. The Unified Vectors team achieved Top 20 in the Team Round.
[…] Problem 9 – Multiplicative Group […]