I have my exam in one week but I am still struggling with number properties. Specifically, questions relating to divisibility, factors, and even/odd. I've been going through the forums looking at threads for assistance but haven't had much luck. I'm wondering if anyone has any tips to help me improve in these areas. Thanks!
Hey Todd,toddg wrote:I have my exam in one week but I am still struggling with number properties. Specifically, questions relating to divisibility, factors, and even/odd. I've been going through the forums looking at threads for assistance but haven't had much luck. I'm wondering if anyone has any tips to help me improve in these areas. Thanks!
I'm new to this site but NOT to GMAT math problems.
I will do my best to offer you some assistance, but it would help to get some specific questions and then work from there. In the meantime, though, here are some key things to keep in mind...
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1. All positive integers can be uniquely represented as a product of prime numbers (raised to integer-value powers); you'd be surpised how many problems actually boil down to this simple fact. Some examples:
2 = 2^1
4 = 2^2
36 = 2^2 * 3^2
Once you break down a positive integer into this form, you can easily figure out the number of ALL factors (both prime and non-prime). To calculate this, you add 1 to each power and multiply them together. Working with the above three examples again:
2 = 2^1. Since the base is to the first power, we add one to it, which gives us 2 distinct factors (1 and 2)
4 = 2^2. This time we have the base to the power of two. We add one to it, which gives us 3 distinct factors (1,2, and 4)
36 = 2^2 * 3^2. For this example, we have the first base and the second base to the power of two, which gives us 3*3 = 9 distinct factors (1,2,3,4,6,9,12,18,36)
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2. I'm not sure exactly what you mean by odd/even, but there are some basic formulas to figure out the sums of consecutive integers, odd/even integers, etc.
Sum(where i equals 1 through n) i = n*(n+1)/2
Sum(where i equals 1 through n) i^2 = n*(n+1)*(2n+1)/6
Sum(where i equals all even integers from 1 through n) = n/2*(n/2+1) {assumes n is even}
Sum(where i equals all odd integers from 1 through n) = ((n+1)/2)^2 {assumes n is odd}
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3. Some divisibility tips and ideas (mod's, remainders, etc)
Determining divisibility by 3, 6, and 9:
a. The remainder of an integer when divided by 3 is the same as the remainder for the sum of its digits when divided by 3.
(ie. 13344 -> sum of digits is 15, which means that it is divisible by 3)
b. To determine whether an integer is divisible by 6, we need to see whether it's divisible by 2 and also divisible by 3, since 6 = 2*3. To do that we check to see whether it's even and also apply the divisibility by 3 check.
(ie. 13344 -> since it's even, it is divisible by 6 because we already know it's divisible by 3
c. As with 3, an integer's remainder when divided by 9 is the same as the remainder of the sum of its digits when divided by 9.
(ie. 13344 -> the sum is 15, which is NOT divisible by 9)
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4. Some more involved remainder ideas that will save you TONS of time for the right types of problems...
(note: 6 mod 5 = 1, mod just means the remainder of 6 divided by 5.)
a. (A + B) mod C = [(A mod C) + (B mod C)] mod C
I call this the additive property of remainders, and it works regardless of how many numbers are added up. Here's an example to make it easier:
9999 mod 5 = [9000 mod 5 + 900 mod 5 + 90 mod 5 + 9 mod 5] mod 5 = 9 mod 5 = 4
b. (A * B) mod C = [(A mod C) * (B mod C)] mod C
I call this the multiplicative property of remainders, which really works just like the additive property. Another example:
What is the remainder of 22*77*98*115 when divided by 5? Keep in mind that as I write this, I have absolutely NO intention of multiplying all of that crap out. Let's see how...
(22*77*98*115) mod 5 = (22 mod 5 * 77 mod 5 * 98 mod 5 * 115 mod 5) mod 5 = (2*2*3*0) mod 5 = 0
c. (A^B) mod C = [(A mod C)^B] mod C
I call this the power property of remainders. This one really saves time when used properly and is actually pretty powerful. An example to show you what I mean:
7^112234 mod 2 = [(7 mod 2)^112234] mod 2 = 1^112234 mod 2 = 1 mod 2 = 1
Hopefully that gets you started.
