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Papgust's GMAT MATH FLASHCARDS directory

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kvcpk Legendary Member
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Post Tue Aug 03, 2010 12:41 am
samark wrote:
Is this rule only applicable for powers of 2? Thanks!
No.. this rule is applicable for any other number.

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blaster Master | Next Rank: 500 Posts
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Post Tue Aug 17, 2010 11:43 pm
Dear papgust, can you please share that flash cards with us in Word document format?

binit7 Newbie | Next Rank: 10 Posts Default Avatar
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Post Sat Sep 18, 2010 12:22 am
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)

Quote:
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.



gmailmba Newbie | Next Rank: 10 Posts Default Avatar
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Post Sun Oct 17, 2010 6:40 am
Thanks papgust! I found your notes very helpful.

JennySpalek Newbie | Next Rank: 10 Posts
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Post Sun Feb 20, 2011 3:37 am
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!

kingluis Junior | Next Rank: 30 Posts Default Avatar
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Post Mon Apr 25, 2011 11:52 am
Thanks for sharing your notes

krishp84 Junior | Next Rank: 30 Posts Default Avatar
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Post Mon Jul 18, 2011 7:27 pm
papgust - Can you correct he information here ?

a^n + b^n:

1. NEVER divisible by a-b

This is WRONG.eg:
3^2+2^2 is divisible by 3-2=1
3^2+6^2 is divisible by 3-6=-3
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

krishp84 Junior | Next Rank: 30 Posts Default Avatar
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Post Mon Jul 18, 2011 7:38 pm
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]

matthewdavid69 Newbie | Next Rank: 10 Posts Default Avatar
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Post Sun Jul 24, 2011 8:45 am
I'm confused. How can n be < -10 AND > 10 ? Should these be negative reciprocals (i.e. 10 > n > -10)?

papgust wrote:
Another point to remember:

Example:

-1/10 < n < 1/10

After taking reciprocal of n, FLIP SIGNS!

-10 > n > 10

herpinderpinthrowaway Newbie | Next Rank: 10 Posts Default Avatar
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Post Tue Nov 08, 2011 8:34 am
Hi, I'm new around here but I really appreciate papgust's effort, so I compiled his posts into a word document. Hope it helps, feel free to correct errors.
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arpitad Junior | Next Rank: 30 Posts
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Post Thu Dec 22, 2011 6:06 pm
Papgust, thanks so much! Your notes are so helpful!

Thanked by: outty
mohhanafy20 Newbie | Next Rank: 10 Posts Default Avatar
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Post Sun May 06, 2012 11:30 pm
you are truly amazing!

noyj91 Newbie | Next Rank: 10 Posts Default Avatar
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Post Thu Mar 07, 2013 6:21 pm
Has anyone managed to put this amazing info in PDF or Word file?

Edit: just found herpinderpinthrowaway 's word doc...thanks!

jitsy Senior | Next Rank: 100 Posts Default Avatar
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Post Tue Jul 09, 2013 4:26 am
Krishp84 (and JennySpalek this should help you as well), I guess you calculated the LCM for both numerators and denominators. The HCF of 3 and 10 (the highest number common in them) is 1 [3=1*3 and 10=1*2*5]. So the answer to the question JennySpalek asked should infact be 4/1 or just 4 (and not 4/30). Hope this helped.


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]

drkomal2000 Newbie | Next Rank: 10 Posts
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Post Mon Feb 24, 2014 10:03 pm
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
Hi Papgust
Thanks a lot for your notes. Can you or anyone from the community give me example of the pattern 1? I know you gave one example but Its confusing for me.

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