Grades 11-12 Video Solutions 2023-2023_11-12

Video Transcription

Problem number 5. We call a positive integer n 2 prime if it has exactly 3 different divisors,  namely 1, 2, and n itself. How many different 2 prime integers are there? Let's say the number n is equal to 2k, because 2 is a factor so we can write it as 2 times some integer k,  and thus n must be divisible by 1, because all integers are divisible by 1, and divisible by n, because all integers are divisible by themselves, and divisible by 2 and k,  because n is equal to 2 times k. For n to only have 3 different factors including 2, in this case 2 should equal k, because k cannot equal n and k cannot equal 1,  so 2 should equal k, and thus n equals 2 times 2 equals 4 is the only 2 prime integer, and so our answer is 1.

Video Summary

The problem defines a "2 prime" integer as a positive integer that has exactly three divisors: 1, 2, and the integer itself. By analyzing the factors of such a number, it is determined that the number must be in the form \( n = 2k \) and satisfy specific conditions to limit its divisors to these three. After examination, it is concluded that the only integer meeting these conditions is 4. Therefore, there is only one 2 prime integer, and the answer is 1.

Keywords

2 prime

integer

divisors

number theory

unique solution