I don't get this math question.

Test Preparation SAT Preparation
1

<p>The question was:</p>

<p>"If n and p are integers greater than 1 and if p is a factor of both n + 3 and n + 10, what is the value of p?"</p>

<p>The answer is 7.
I looked on the Collegeboard website and their explanation was:</p>

<p>"Since p is a factor of n + 3 and n + 10, it follows that p is also a factor of (n + 10) - (n + 3) = 7."</p>

<p>Why does it follow that p is also a factor of 7? Could someone explain this to me? Or just give me an example in which this would work? D:</p>

2

<p>I’m really bad at explaining things to people, but I’ll give it a try.</p>

<p>See if this helps: Say 4 is a factor of both 16 and 12, then it would follow that 4 is a factor of 16-12, which equals 4. 4 is a factor of 4.</p>

<p>Since 9 is a factor of both 81 and 54, it follows that 9 is a factor of 81-54, which is 27. 9 indeed is a factor of 27. </p>

<p>You get try this with any combination of numbers greater than 1 and it should work out.</p>

3

<p>Thanks, that helped a lot. :D</p>

<p>Just wondering, is this like a theorem or something? Because I never learned this.</p>

4

<p>i would prove it using number theory but it can get a little complicated :p</p>

5

<p>

</p>

<p>No, I don’t think it’s a theorem either. Like other SAT math problems in general, there’s no specific theorem or formula and you kind of just have to figure out a rule to solve the problem yourself:)</p>

6

<p>A lot of the Math SAT is plugging in numbers until you get the right answer, but you have to be sure, because it still could be wrong, but 7 is definitely right for this one.</p>

7

<p>for questions like this, plugging in works best, but sometimes plugging in can backfire. Sometimes for questions like this, I try to find patterns which might help me. As you read in xrCalico23’s post, the difference of two factors of a certain number is a factor of that certain number also. So after i see the question, I might think, “12 and 4 are factors of 2, and their difference, 8, is also a factor of 2.” but i’ll need my thought to be solid so i’ll try different things like odd numbers and non-integers and then come to the conclusion that the difference of the factors of a number is a factor also. </p>

<p>It might be hard to do this for this question but it helped me before on similar questions.</p>

8

<p>n = 4
p = 7</p>

<p>I remember this question from a BB practice test. Just plug in until you get a value for n that fits.</p>

9

<p>“If n and p are integers greater than 1 and if p is a factor of both n + 3 and n + 10, what is the value of p?”</p>

<p>n + 3 = 0 (mod p)
n + 10 = 0 (mod p)
n + 10 = n + 3
7 = 0 (mod p)
p = 7</p>

<p><a href=“http://en.wikipedia.org/wiki/Modulo_operation[/url]”>http://en.wikipedia.org/wiki/Modulo_operation&lt;/a&gt;&lt;/p&gt;

10

<p>what were the choices? because you can immediately cross out any even ones because one of them, (n+3) or (n+10) has to be odd and the other even.</p>

11

<p>n + 3 = 0 (mod p) – here you mean that when (n+3) and 0 are divided by “p”, they both have the same remainder, 0. Same goes with “n + 10 = 0 (mod p)”.
but then how could you conclude,
n + 10 = n + 3
7 = 0 (mod p)?</p>

12

<p>The choices were
3
7
10
13
30</p>

<p>And I’m going through the replies. Thanks guys. (:</p>

13

<p>This is a suggestion that might be useful in solving other problems, strawbearries:</p>

<p>p is a factor of n + 3 means n + 3 = ap, where a is some integer.
p is a factor of n + 10 means n + 10 = bp, where b is some integer.
Look at the two equations
n + 3 = ap
n + 10 = bp
At this point, you have two equations in 3 unknowns (a, b, and p). So you will not be able to solve them, but you could make some progress toward a solution by subtracting the upper equation from the lower one. This gives
7 = (b-a)p
We know that b and a are both integers, and b must be greater than a, because a p = n +3 while b p = n+10. Therefore b-a is a positive integer, so p is a factor of 7.</p>

<p>Bassir’s approach is elegant, but I’m guessing that it is not especially useful to you, if you are not familiar with modular arithmetic.</p>

14

<p>Thank you sooo much QuantMech!
Your way seems most like a way that the SAT writers would want someone solve a problem.</p>