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# Question of the Week for January 1

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Thanks for joining us here. Let's work through this!

We could start by thinking of values of b and c that satisfy the inequality b ≥ c > 2, Since we know b and c are integers and that we're looking for the smallest possible sum of b and c (b + c), we might imagine the smallest integers that fit the inequality:

c = 3 and b = 3

However, we would need to see if these guys work in the equation 7(b - 2) = 17(c - 2)

Does 7(3 - 2) = 17(3 - 2)? No, clearly it does not. The left hand side is way too small.

Thus we know that we would need different values for b and c. How could we go about finding values for b and c that work?

Let's start by ignoring b and c for a second. What's going on in the equation?

On the left hand side we have 7 times a number and on the right hand side we have 17 times a number. In other words, we're going to have to multiply 7 and 17 by different integers to get the two sides equal to each other. How could 7 times something equal 17 times something?

Think about it. We'd need to find a common multiple of 17 and 7. In fact, since we want the least possible value of b + c, we'd want the least common multiple of 17 and 7. What is this?

17 + 17 = 34 Is this a multiple of 7? No.

17 + 17 + 17 = 51 Is this a multiple of 7? No.

Do you see the pattern? 17 ∙ 2, 17 ∙ 3, etc.

What will be the least common multiple of 17 and 7? Since they are both prime, the least common multiple of 17 and 7 will be 17 ∙ 7.

17 ∙ 7 = 119

Thus, we need the equation to work like this:

7(b - 2) = 17(c - 2) = 119

Let's solve for b and c separately:

7(b - 2) = 119

b - 2 = 17

b = 19

17(c - 2) = 119

c - 2 = 7

c = 9

b + c = 28

So you got it! It's (E).

But wait there's more!

Is there any easier way to do this problem. Remember to Analyze the Answer Choices™. What's going on in those answers?

A ) 4 is pretty clearly impossible. The smallest possible values for b and c are 3 and 3. So, the smallest sum is 6.

B ) Not much to do with this one. Leave it in as a possibility.

C ) 14 is just 7 times 2. This is likely a trap answer. Eliminate it.

D ) 24 is just 17 + 7. This is likely a trap answer. Eliminate it.

E ) Not much to do with this one either. Leave it in as a possibility.

Your choices are B and E. You now have a 50/50 shot of getting it right with no work. I like those odds!