This is a beautiful problem from ISI MSTAT 2019 PSA problem 11 based on multiplication principles. We provide sequential hints so that you can try.
How many positive divisors of \( 2^55^3{11}^4 \) are perfect squares?
Basic counting principles
Divisors of a number
Answer: is 18
ISI MStat 2019 PSA Problem 11
A First Course in Probability by Sheldon Ross
See in order to get the positive divisors of \( 2^55^3{11}^4 \) that are perfect squares , we need to take only the even powers of the primes {2,5,11}.
Now the maximum powers of 2 is 5 . So there are 3 choices for even powers {0,2,4} .
The maximum powers of 5 is 3 . So there are 2 choices for even powers {0,2}.
Now the maximum powers of 2 is 5 . So there are 3 choices for even powers {0,2,4} .
The maximum powers of 11 is 4 . So there are 3 choices for even powers {0,2,4}.
Hence by multiplication principle we have in total \( 3 \times 2 \times 3 =18 \) such positive divisors.
Publishing original scientific research in a peer-reviewed journal is an achievement many scholars work toward for years. Achieving this feat early in one's academic journey requires exceptional intellectual rigor, creativity, and persistence. Aaditya Punatar and Muskan Sadana have reached this remarkable milestone, successfully publishing their collaborative research paper in the prestigious Journal of Emerging Investigators […]
Her project on quantum-accelerated Steiner trees for 15-minute cities earned a place among the finalists at the world's largest pre-college STEM competition. Prishaa Shrimali of Marymount School of New York, was selected as a Grand Finalist at the 2026 Regeneron International Science and Engineering Fair (ISEF) in Phoenix, Arizona. Her project was Optimizing Urban Accessibility: […]
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 […]
