Grades 7-8 Video Solutions 2025-2025_7-8

2025_7-8_16

Video Transcription

Problem number 16. Julia wants to fill in each box with a different prime number less than 20 so that the value of a is an integer. What is the maximum value of a?  To solve this problem, let's list out the prime numbers less than 20. And in particular, you should remember that a prime number is a number that's only divisible  by 1 and itself. In this case, the primes that are less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19.  Now, notice that there are only 8 of those numbers. And in the picture, we only see 8 boxes. And we also know that each box has to have a different prime number.  Since there are only 8 different primes, that means we have to use each of these numbers  exactly once. We can't remove any of these numbers because we only have 8 numbers and 8 boxes.  This means that we can start to figure out what the numerator and denominator will be. The numerator, since it has 7 of the primes, will be the sum of all the primes  minus whatever number is in the denominator, which I've been calling the unused number, since that number is not used in the numerator. And the denominator will just be that unused  number. That is, we can figure out the value of a just by looking at what number is in the  denominator. Since if we know that number, then all the other numbers will go to the numerator,  and we can calculate the value of a. The sum of all the numbers is 2 plus 3 plus and so on  until 19. If we compute that sum, we will get 77. So now let's figure out what the value of a could be. Let's call the unused number, the number that goes to the denominator,  x. That means that the value of a will be 77 minus x, all divided by x, because the numerator  will be 77 minus x, since we're not including x in the numerator, and the denominator will just  be x, since that's what the denominator is defined to be. That is, we have to find the value of x  such that 77 minus x all over x is a whole number. We can simplify this expression by dividing out  that 77 minus x all over x is equal to 77 over x minus x over x, and x over x is always equal to  1. So this value is just equal to 77 over x minus 1. Now the question is, what values of x  make this a whole number? And in particular, we want this value to be as large as possible.  If we think about what factors of 77 there are, the only values that we can see in our list of  primes less than 20 are 7 and 11. Those are the only factors of 77. Since we want this value to  be as large as possible, to maximize it, we should select x equals 7, because that will make the  division a larger result. If we instead select x equals 11, then the dividing number will be a lot  smaller, and so the result, the quotient, will be a lot smaller. So this means that to maximize,  we should pick x equals 7. In that case, the value is 77 divided by 7 minus 1, which is equal  to 10. So just to recap, the value of x that we selected has to be a divisor of 77, and the only  divisors of 77 that we see in the list are 7 and 11. x equals 7 is better because the quotient  will be larger. In that case, the quantity that we get for a is 10, so the answer is c, 10.

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