Math is so beautiful. Sorry guys, I am struggling with the GMAT too. Largely on the numbers properties which is why I'm here but when I think of the explanations I am just amazed by people whose brains can unravel these puzzles. It is awesome. I don't know if it is a matter of nurture vs. nature, I bet it is a bit of both but I guess beat the gmat is about the idea that nurture can work depending on how long you want to work. Sorry for the preaching but felt the need to say this. I've been at this for 3 months with hours and hours of doing these problems and after fixing one problem I'm finding other problems. My score is pretty much the same even though I understand the answers more and I've come to the conclusion that it is based on these numbers problems that the test administrators keep saying is worth 600 - 700. I solve 800 non-number properties problems easier than this. The issue is getting your head around these concepts. So I'm looking for as many sample numbers properties questions as I can get if anyone has it.
N e case, an explanation of the answer is this:
The multiples of this function IS every odd integer under 50 because every odd integer under 50 multiplied by 2 is an even integer under 100; e.g., 49*2 = 98 and 47*2=95, etc. This is what is meant by divide all the even factors by 2. 98/2 = 49; 94/2 = 47. More accurately stated, it is divide every other even number starting with 98 by 2 because the numbers in between are 2 multiplied by another even number (96/2 = 48); but that explanation would probably be confusing.
51*2= 102. 102 is greater than 100 (the question says, the function is the product of every even integer from 2 to n and we are told that n is 100); therefore 102 cannot be one of the multiples of this function and any odd number above 50 doesn't have to be considered.
Now you know that every odd number below 50 is a factor in the final product of the function. You also know that all prime numbers except for 2 are odd. So that means you can now consider what the possibility is that P, the prime number is one of those numbers under 50.
Well, we know all the prime numbers under 50 starting with 2, 3, 5, 7, 11, etc. Since the function is h(100) + 1 and we know that P is the lowest prime that divides h(100) + 1 evenly, we know that 3, 5, 7, 11 ... will not divide evenly into h(100) +1 because IF h(100) is a multiple of the prime then the next multiple of the prime will include an addition of that prime.
For example, 14 is a multiple of 7 the next multiple of 7 is 14 + 7 or 21. Similarly, 6 is a multiple of 3 the next multiple of 3 is 6 + 3 or 9. So that means that if h(100) were 14 then h(100) + 1 = 15, which is not a multiple of 7. Or if h(100) were 6 then h(100) + 1 would be 7, which is not a multiple of 3. Therefore, we know that P can't be any of the prime numbers under 50 and it must be one that is above 50.
Remember that this is a very large number - the product of every even integer from 2 - 100. There is a possibility that there is a prime above 50 that we don't want to and don't have to calculate above 40 that is a factor of this large number.
I hoped this helped. Explaining helps you learn too so this was good for me.
