samark wrote: Is this rule only applicable for powers of 2? Thanks!

papgust wrote:How to test whether a number is prime or composite:

Before we start off, what is a prime number and a composite number? (For people who are not sure)

A Prime Number is a positive integer that is divisible by ONLY 2 numbers (1 and itself). Whereas, A composite number is a positive integer which has divisor(s) other than the 2 numbers (1 and itself).

Ok, coming back to the point. I will name the number as n for simplicity. Following are the steps to test whether a number is a prime or composite,

1. Identify the perfect square (P.S) closest to the n.
2. Compute the square root of P.S
3. List all prime numbers upto the computed square root
4. Check if all listed prime numbers divide n equally. If not, then n is a prime. Even if atleast one of the listed prime numbers divide n, then n is a composite.


Example:

Take n as 113. To test whether 113 is a prime,

1. 100 is the closest perfect square to 113 (Remember that you take a closest perfect square that is smaller than n itself!)
2. Square root of 100 ==> 10
3. Prime numbers upto the square root (10) ==> 2,3,5,7.
4. Check whether 2,3,5,7 divides 113. None of the numbers divide 113. So, 113 is a prime.

For a number to be prime:

All the prime numbers > 3 will be of the format 6N+1 or 6N-1.
As, all the prime numbers >3 give a remainder of 1 or 5 when divided by 6.

Take the previous example:
113 = (6 * 19) - 1. Hence,it is prime.

papgust wrote:Simple Facts:

I have not covered the rest of the topics you posted.
Will let all know if there are discrepancies.

a^n - b^n:

1. ALWAYS divisible by a-b
2. If n is even, it is divisible by a+b
3. If n is odd, it is NOT divisible by a+b


a^n + b^n:

1. NEVER divisible by a-b
2. If n is odd, it is divisible by a+b
3. If n is even, it is NOT divisible by a+b

JennySpalek wrote:Dear Papgust, you are truly amazing!
Thank you so much for sharing your notes.

I have just a question regarding "how to calculate LCM and HCF of fractions"
Could you please be so kind as to give an example of both?

I could not really think of an example. How about 2/3 + 4/10? For the LCM of 2 Fractions for instance, you said it is the LCM of Numerators / HCF of Denominators, but in this example (3= 1*3, 10= 2*5, they do not have any common factors!). So you cannot apply it then? I mean of course you could just easily convert to 20/30 + 12/30, but I just wanted to "apply" your formula, and got confused!

Thanks!

papgust wrote:Another point to remember:

Example:

-1/10 < n < 1/10

After taking reciprocal of n, FLIP SIGNS!

-10 > n > 10

krishp84 wrote:

JennySpalek wrote:Dear Papgust, you are truly amazing!
Thank you so much for sharing your notes.

I have just a question regarding "how to calculate LCM and HCF of fractions"
Could you please be so kind as to give an example of both?

I could not really think of an example. How about 2/3 + 4/10? For the LCM of 2 Fractions for instance, you said it is the LCM of Numerators / HCF of Denominators, but in this example (3= 1*3, 10= 2*5, they do not have any common factors!). So you cannot apply it then? I mean of course you could just easily convert to 20/30 + 12/30, but I just wanted to "apply" your formula, and got confused!

Thanks!

Jen - Not sure if you or anyone is even following the posts...
Let me rephrase your question -You want to find the LCM of 2/3,4/10 (not 2/3+4/10 because this is a single number)
If you are calculating LCM of 2/3,4/10
LCM(2,4)=LCM(2,2x2)=2x2=4
HCF(3,10)=3x10=30(because as you mentioned there are no common factors)
So, LCM(2/3,4/10)=LCM(2,4)/HCF(3,10) = 4/30 = 2/15

Same concept applies the other way when calculating HCF/GCD
Hope this helps you/anyone.[/b]

papgust wrote:Warning: Some people may not find this approach comfortable. Some may find it comfortable. Please follow and practice only if you are comfortable with this approach. Otherwise, please ignore it.


Sometimes, we get one type of question in GMAT where we need to calculate units digit of integers raised to some power. I found a shortcut where you could save time by remembering some patterns.

How to find unit digit of powers of numbers:

Pattern 1:
Unit's place that has digits - 2/3/7/8

Then, unit's digit repeats every 4th value. Divide the power (or index) by 4.

After dividing,
If remainder is 1, unit digit of number raised to the power 1.
If remainder is 2, unit digit of number raised to the power 2.
If remainder is 3, unit digit of number raised to the power 3.
If remainder is 0, unit digit of number raised to the power 4.

Pattern 2:
Unit's place that has digits - 0/1/5/6

Then, all powers of the number have same digit as unit's place.

For e.g., 6^1 = 6, 6^2 = 36, 6^3 = 216, 6^4 = 1296


Pattern 3:
Unit's place that has digit - 4

Then,
If power is odd --> unit's digit will be '4'
If power is even --> unit's digit will be '6'

Similarly,
Unit's place that has digit - 9

Then,
If power is odd --> unit's digit will be '9
If power is even --> unit's digit will be '1'


Example:
Let's take a long number - 122 ^ 94. Find unit's digit.

Unit's place is 2. So, it repeats every 4th term of the power.
So, divide the power by 4. 94 % 4 ==> 2 (remainder).

Raise the unit digit of the base number to the power (2 - remainder). 2^2 = 4.

Thus, 4 is the unit's digit of 122^94.


I found this approach very easy and comfortable. So, see how comfortable it is for you and apply.


Real GMAT Problem: OG-12 PS #190