A Fraction and a Prime Number Question
Date: 9 Feb 1995 15:07:05 -0500 From: Pat Couch Subject: Kevin's Question Dear Dr. Math, I have two questions I would like some help with. Here is the first question: If 1/4 of a number equals 1/20, then 5/4 of this same number equals a) 1/4 b 1/5 c) 1/16 d) 1/64 I'm not sure how to figure out the answer. 2. The product of the first 100 prime numbers is not divisible by: a) 44 b) 55 c) 66 d) 77 Could you please help me with this one too. Thanks. Kevin Richardson
Date: 12 Feb 1995 02:13:53 -0500 From: Dr. Ken Subject: Re: Kevin's Question Hello there! Well, here's what we can do. Let's translate your first sentence into an algebraic statement (that just means we'll let letters stand for numbers, basically). If we let x represent the number in question, we get 1 1 --- x X = ---- 4 20 Then we think about what we're trying to find out. We're trying to find a value for the quantity (5/4) times X, right? So what happens when we multiply the equation by 5? We get 5 5 5 1 --- x X = ---- = ------- = --- 4 20 4 x 5 4 So that's our answer, 1/4. Did this make sense? If you still don't understand, write back to us. This problem is a little more interesting. Let's look at a list of the first few prime numbers: 2, 3, 5, 7, 11, 13, 17, 23, 29, ... Now, it looks at first like we're going to have to find out what the product of the first 100 prime numbers is, and then divide it by the four options we're given. But this seems too hard, so there must be some kind of trick. And in fact, there is. What do you notice about the list of primes there? How many of them are even numbers (divisible by 2)? Will any prime number except 2 be even? The answer is no, and I'll leave it up to you to figure out why (hint: what's the definition of a prime number?). Now look at the first option we're given: 44. If a number is divisible by 44, that means it's divisible by 4 and 11, since 4 x 11 = 44. But if we mulitiply any number of prime numbers together, the only one that's going to give us any twos is 2, because it's the only even number among the primes. So we couldn't possibly get _two_ twos (i.e. 4) by only multiplying together prime numbers. I hope this helps you understand these problems. They're neat once you understand them! -Ken "Dr." Math
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