Grades 11-12 Video Solutions 2010-11&12 Video Solutions 2010 problem15

11&12 Video Solutions 2010 problem15

Video Transcription

Video Summary

The question asks how many three-digit prime numbers \( p \) satisfy that \( p - 1 \) has exactly one prime divisor. Since \( p - 1 \) is even, it must be a power of 2 for it to have only one prime divisor, 2. Three-digit numbers plus 1 that are powers of 2 are \( 2^9=512 \), \( 2^8=256 \), and \( 2^7=128 \). Only \( p-1=256 \) (\( p=257 \)) results in a prime number. Thus, 257 is the only three-digit prime fulfilling the condition. Therefore, the answer is one such number.

Keywords

three-digit prime

prime divisor

power of 2

prime number

257