Grades 9-10 Video Solutions 2023-2023_9-10

Video Transcription

Tim calculates the mean of 5 different prime numbers. His answer is an integer. What is the smallest possible integer he could have obtained?  So we want to minimize the mean, which we know is just the sum of the 5 numbers divided by 5.  Therefore, we want to minimize each of the 5 individual prime numbers. The easiest way to do this would be to take the 5 smallest prime numbers.  We know that the smallest 5 prime numbers are 2, 3, 5, 7, and 11, which together add to 28. However, this isn't a multiple of 5, and 28 divided by 5 won't be an integer.  Then, we try to remove one of these integers and instead use the next biggest prime possible.  So, the next biggest prime is 13. To minimize our sum, we want to change the biggest number we're already using for the number we're replacing it with.  So we try to replace 11 with 13. When we replace 11 with 13, we notice that our sum of the 5 primes is now 30.  30 is a multiple of 5 and will give an integer mean. We know that 30 divided by 5 is 6, and therefore our correct answer is C, 6.

Video Summary

Tim minimizes the mean of five different prime numbers by selecting the smallest primes possible. Initially, the smallest primes are 2, 3, 5, 7, and 11, but their sum, 28, isn’t divisible by 5. By replacing the largest prime, 11, with the next largest prime, 13, the sum becomes 30. This sum, 30, is divisible by 5, resulting in a mean of 6. Thus, the smallest possible integer mean using five different prime numbers is 6.

Keywords

prime numbers

mean

smallest primes

integer mean

sum divisible