<p>I've used that title before but anyways. I'm in test 9, section 8 problem 14 on page 919 of the Blue Book and I was wondering if you could help me out.</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>While doing this section I was so clueless so I just picked 4 (random number) to substitute in for n and it ended up giving me the right answer. But I still want to understand how to do it without luck.</p>
<p>I would do it with luck…
lol</p>
<p>Just to make sure, is the value of p 7?</p>
<p>Since the difference between n+10 and n+3 is 7, for some number n, both n+3 & n+10 are bound to be multiples of 7. Therefore, you can assume with relative (to your case) confidence that p is 7.</p>
<p>A more sure fire way to find the answer is to start plugging in integers greater than 1,
n= x, n+3, n+10
n=2, 5, 12
n=3, 6, 13
n=4, 7, 14
n=5, 8, 15
n=6, 9, 16
n=7, 10, 17
So far, the only pair of n+3 and n+10 that has a common factor is 7 & 14, where p is 7; if you continue,
n=8, 11, 18
n=9, 12, 19
n=10,13, 20
n=11, 14, 21
Here, we can see the ones digits starting to repeat, & we also see that at n=11, 14 & 21 have a factor of 7. This would mean p=7</p>
<p>To solve these types of problems, you need to use reasoning to try to get to the answer. The second way is how I came about the answer when I took that test.</p>
<p>p is a factor on n+3. That means n+3 = Ap where A is an integer.
p is a factor of n+10. that means n+10 = Bp where B is an integer.</p>
<p>Subtract n+3 from n+10, and you get: (B-A)p = 7. Since 7 is a prime number this is only possible if either p=1 or p=7. It can’t be 1 since in the problem statement p is greater than 1.</p>
<p>Ok, I can’t be certain as to the “proper approach” but I can tell you how to solve it with logic. Lets make up an arbitrary number; x. X represents the value of n+10. Another number, y, represents the value of n+3. The difference between x and y is only 7 though we do not know P still. However, since P must factor into both X and Y, its maximum value may only be 7 and since since 7 has no real integer factors, it can only be 7. Hopefully you can follow that.
Basically the difference between n+10 and n+3 is always 7 regardless of how big n is. Therefore, P may only be equal to a factor of 7 and the only real numbers that exist are 1 and 7.</p>
<p>oops 2 people answered in the time I was making this post >.<</p>
<p>Ohh sorry, I forgot to put down the choices (knew I was missing something) the choices are 3,7,10,13,30 and yes the answer is 7. Thanks you guys really helped :)</p>