Try this problem:
If n is a multiple of 5 and n = p 2 q , where p and q are prime numbers, which of the following must be a multiple of 25?”
(A) p 2
(B) q 2
(C) p q
(D) p2q2
(E) p3q“
So p and q are primes. They could be 2, 3, 5, 7 or so on. It doesn’t say that p and q are different prime numbers, so they could also be the same number. And I’m going to use some theory here: if p2q equals n and n is a multiple of 5, then that 5 must be contained in either p or q. And since those two numbers are primes, either p or q is 5. (Or maybe both are!)
How can you figure that out? Look at this (totally different) example:
56 = 8 × 7
56 is a multiple of 4. Do either of the two numbers on the right-hand side contain a 4? Yes, 8 contains a 4.
Come up with an equation like the one above in which the number on the left is a multiple of some number, but you CANNOT find that number in one of the numbers on the right.
56 = 7 × [something that does NOT contain a 4]
We can’t do it. If the two sides are equal, they must simplify down to the same primes:
56 = 2 × 2 × 2 × 7
8 × 7 = 2 × 2 × 2 × 7
Okay, so that’s how you know that if the n, on the left side, contains a 5 as a factor, then the p2q on the right side also has to contain a 5 somewhere. Since we were told that p and q are both primes, we know that at least one is equal to 5.
Which one? We have no idea. Now, the problem is asking us how to get 25. Well, if p is 5, then we’d need p2 to get 25. And if q is 5, then we’d need q2 to get 25.
So how can we guarantee that we get 25? Have both of those! With p2q2, it doesn’t matter whether the p is 5 or the q is 5 “ either way, we square whichever one is 5 to get 25!
The correct answer is D.
The key takeaway here is this: when we’re told two expressions equal each other, then those two expressions must simplify down to the same pieces (on each side of the equation). Remember that.
Ready for the next one?
If n and y are positive integers and 450y = n3, which of the following must be an integer?”
(A) None
(B) I only
(C) II only
(D) III only
(E) I, II, and III
We’re told that we have positive integers and there’s an equation. Now, what’s the question asking? Which of the following must be an integer okay, and then each of the roman numerals contains y divided by some primes oh, I get it! If doing the division results in an integer, then that means y is divisible by those primes. This is actually a divisibility question, too, even though they don’t say that word in the problem! Also, this is a MUST be true question, just like the last one.
So, what do we do with the equation, again, on divisibility questions? Ah yes “ that means the left-hand side must ultimately break down to the same thing on the right-hand side. Let’s see if that helps us at all on this one.
450 = 45*10 = 3 × 3 × 5 × 2 × 5
n3 = n × n × n
Hmm. So, here’s what we’ve got (I’m going to rearrange the numbers that make up 450 to put them in order):
2 × 3 × 3 × 5 × 5 × y = n × n × n
Now what? Hmm, what’s the significance of having three n‘s on the right-hand side? Each n is the same number and the n‘s on the right have to contain all of the same pieces as the stuff on the left
That’s interesting. Each n has to contain the exact same thing, and I can’t break primes down any further. So if there’s one 2 on the left, there actually has to be at least three 2’s, because each n needs to get its own 2.
Too theoretical? Try a simpler, real-numbers-only example.
2 × 2 × 2 × 5 × 5 × 5 = b × b × b
What is the value of just one single b? We just split each set of 3 numbers on the right “ one 2 for each n and one 5 for each n, or n = 2 × 5.
In our harder problem, with the y, we don’t have all of the numbers on the left because we’ve got this variable y. So we basically have to infer that, if there’s one 2, there must be at least three 2’s (so that each n can have its own 2). By the same token, if there are two 3’s, there actually must be at least three 3’s, and ditto for the 5’s.
Where are those other numbers? They have to be part of the y variable. So the y has to contain at least two 2’s, one 3, and one 5. (Technically, it could contain even more “ maybe it contains three 17’s as well! But the question asks us what must be true, so we don’t have to speculate about what else we might have.)
If the y must contain two 2’s, one 3, and one 5, then how can we use that knowledge to evaluate the three roman numerals? First, let’s represent y mathematically:
y = 2 × 2 × 3 × 5 × ?
We know we have those 4 numbers plus there might be other stuff, so we’ll include the question mark, just in case.
If I take that and plug it into the first roman numeral, am I going to get an integer?
Let’s see:
Start crossing stuff off. You can definitely eliminate everything on the bottom! That means you don’t have anything left on the denominator except for the number 1, so this must simplify to an integer. 1 works.
Eliminate answers A, C, and D. Excellent! Notice that we only have to test one more roman numeral, either II or III. That’ll be enough to get the answer, because it can only be either B or E at this point. If one of those roman numerals looks a lot easier to you than the other, make sure to do the easier one next.
Let’s try II.
I can cross off the 2 and the 5. I can also cross off one of the 3s. Can I cross off the other 3? Maybe “ maybe the ? contains another 3 “ but I don’t know for sure. So there’s a possibility that I’m left with a 3 on the bottom of this fraction which means I won’t get an integer. This one isn’t a MUST be true statement, so I can eliminate answer E.
The correct answer is B.
Let’s look at roman numeral III to be thorough (though on the real test, don’t do more work than you have to do!).
This one is like II. I know You can definitely divide out the 2, the 3, and one of the 5’s on the denominator, but you don’t know that you can definitely divide out both 5s. This one is also not a MUST be true statement.
Key Takeaways for Divisibility Problems:
(1) If the problem is about divisibility and we’re given an equation, then we’re probably going to need to break both sides of the equations down to the basic components and compare.
(2) A divisibility problem might not use normal divisibility language (divisible, multiple, factor, ). For example, it may just ask us whether we get an integer for some math setup that includes division “ we have to recognize that they really are asking about divisibility and factors.
(3) On MUST be true questions, we have to find something that is always true, not just sometimes true. There will be traps revolving around things that could be true but don’t absolutely have to be true “ watch out for those traps!
