Grades 9-10 Video Solutions 2024-2024_9-10

Video Transcription

Ali wants to remove some of the integers from 1 to 25 and then separate the remaining numbers into two groups so that the products of the integers in each group are equal.  What is the smallest number of integers Ali could remove? Note that the prime factorizations of the products of the numbers of both groups have  to be equal in order to make the products equal. Therefore, prime numbers that have only one multiple among the numbers from 1 to 25 must be removed.  This includes the numbers 13, 17, 19, and 23. Note also that writing out the prime factorization tells us that 7 has three multiples among numbers from 1 to 25.  At least one must be removed, and it can be found by checking that by removing 7, we can allow the remaining 20 numbers to be split into two groups with equal product, and therefore  our answer is B, 5.

Video Summary

To create two groups with equal product from integers 1 to 25, Ali must remove numbers whose prime factors don’t allow equal division. She must remove numbers 13, 17, 19, and 23, as they appear only once from 1 to 25. Additionally, removing one multiple of 7 (since there are three: 7, 14, and 21) enables a successful division between groups. Thus, by removing a total of five numbers (13, 17, 19, 23, and one multiple of 7), Ali can achieve this partitioning. Hence, the smallest number of integers Ali could remove is five.

Keywords

integer partition

prime factors

equal product

number removal

Ali's strategy