Lesson: Prime Numbers

Comment on Prime Numbers

Hi Brent,

Do you have the video slides for this lesson available for download? I know you have some available in linkedin.

Thanks!
gmat-admin's picture

Hi bcc123,

We don't have slides for each individual video lesson. However, we do have slides that cover all of GMAT math here: https://www.slideshare.net/GMATPrepNow_free/gmat-math-flashcards and here: https://gmatclub.com/forum/new-resource-interactive-flashcards-for-gmat-...

sir doubt
https://gmatclub.com/forum/if-x-y-y-1-and-y-is-a-prime-number-less-than-11-which-of-the-f-223453.html
gmat-admin's picture

Happy to help!

My step-by-step solution can be found here: https://gmatclub.com/forum/if-x-y-y-1-and-y-is-a-prime-number-less-than-...

Cheers,
Brent

Is the positive integer x even?

(1) (x - 1) is a prime number
(2) (x^2 - 1) is a prime number

statement 1 = if x = 3 then x-1 is prime
if x = 6 then x-1 is prime
so insuff
statement 2
x^2-1 = prime
only 2 satisfies this
so suff
is this approach correct???
gmat-admin's picture

Perfect approach!

Cheers,
Brent

Hi Brent, could you please help me with the below question

https://gmatclub.com/forum/how-many-positive-integers-less-than-30-are-either-a-multiple-of-2-an-127362.html
gmat-admin's picture

Hi Brent here's another one I need your help with

https://gmatclub.com/forum/set-s-consists-of-more-than-two-integers-are-all-the-numbers-in-set-s-152717.html

https://gmatclub.com/forum/if-p-is-a-positive-integer-is-p-a-prime-number-172237.html
sir in statement 1 how to know which values to test?
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-p-is-a-positive-integer-is-p-a-prime-numbe...

In this question, we're asked whether p is a prime number.
The GMAT loves to test whether students are aware that 2 is a prime number (in fact 2 is the ONLY prime number that's even). In fact, 2, 3 are the ONLY two consecutive primes.
So, that's why I tested p = 2

Notice that 2 and 3 (aka, p and p+1) have TWO factors each (making them both prime)
So, at that point, I started looking for 2 consecutive integers that each have FOUR factors each, which would make those values NOT prime numbers (aka composite numbers).

Do 3 & 4 work? No, 3 has two factors, and 4 has three factors.
Do 4 & 5 work? No, 4 has three factors, and 5 has two factors.
Do 5 & 6 work? No, 5 has two factors, and 6 has four factors.
.
.
.
Do 14 & 15 work? YES! 14 has FOUR factors, and 15 has FOUR factors.

Cheers,
Brent

https://gmatclub.com/forum/how-many-positive-integers-less-than-30-are-either-a-multiple-of-2-an-127362.html

I'm not sure if the approach I used is right:

Number of multiples of 2 between 2 and 30:

2(1) ... 2(15) 15 - 1 + 1 = 15
Take away 1 as its integers below 30
So total is 14

Number of odd prime numbers below 30:

9

Sum of positive multiples of 2 and odd prime numbers:

2 + 19 = 21
2 + 23 = 25
4 + 23 = 27

Any other calculations would have resulted in overlap and repetition. So total numbers that satisfied the statement was: 14+9+3 = 26

Where am I going wrong and how could this have been done much much more simpler in under 2 mins?
gmat-admin's picture

In your approach, what's the rationale behind choosing the smallest odd prime 3 as opposed to 5 or 7?
gmat-admin's picture

Choosing 3 ensures that I don't miss any values.

That said, in this particular example, I could have chosen 5 or 7 and still reached the correct answer, but it's still best to start with the smallest odd prime.

For example, if I had chosen 11, then I would have missed the opportunity to get 9 as one of the possible values.

Cheers,
Brent

https://gmatclub.com/forum/how-many-positive-integers-less-than-30-are-either-a-multiple-of-2-an-127362.html
i did not get the 30 second approach given by Bunuel
gmat-admin's picture

Question link: https://gmatclub.com/forum/how-many-positive-integers-less-than-30-are-e...

Bunuel's solution is pretty much the same as my 30-second solution (at https://gmatclub.com/forum/how-many-positive-integers-less-than-30-are-e...). The main difference is that Bunuel uses fewer words :-)

Cheers,
Brent

https://gmatclub.com/forum/if-the-integer-n-is-greater-than-1-is-n-equal-to-94720.html

would you please give your approch?
gmat-admin's picture

The product of all the prime numbers less than 20 is closest to which of the following powers of 10 ?
(A) 10^9
(B) 10^8
(C) 10^7
(D) 10^6
(E) 10^5
The answer is D right?

Prime numbers are 2,3,5,7,11,13,17,19.

To approximate the product of those prime numbers we can do the following:
2*5=10
3*7=20
11*19=200
13*17=200

Now lets multiply 10*20*200*200=8000000 and according to scientific notation the result should be written as 8*10^6.

If to take calculator and make a precise calculation the answer is also 10^6. Do I understand it correctly? Please help me out. Thank you.
gmat-admin's picture

Your approach is great.

However, you need to recognize that your approximations are all a bit smaller than than the actual products.
For example, 11 x 19 = 209 (not 200), and 13 x 17 = 221 (not 200) etc.
So, your product of 8 x 10^6 is a bit LESS THAN the actual answer.
If we round UP, we get: 8 x 10^6 ≈ 10 x 10^6 ≈ 10^7

Here's my full solution: https://gmatclub.com/forum/the-product-of-all-the-prime-numbers-less-tha...

Does that help?

Cheers,
Brent

Question link: https://gmatclub.com/forum/set-s-consists-of-more-than-two-integers-are-all-the-numbers-in-set-s-152717.html

Hi Brent, I don't see how this explanation holds good for a set with, say, 4 or 5 numbers. Could you please help on this?

Thanks!
Kashaf
gmat-admin's picture

Question link: https://gmatclub.com/forum/set-s-consists-of-more-than-two-integers-are-...

This is a crazy tricky question!!

The solution wouldn't hold up if there were 4 or 5 numbers. However, since we're not told how many numbers there are, it's possible that there are 3 numbers (if we knew there were 4 or 5 numbers in the set, statement 1 would be sufficient. So, the only way statement 1 is not sufficient is when there are only 3 numbers in the set).

However, when we combine the two statements, we can be certain that all of the numbers are negative.

Does that help?

Cheers,
Brent

Was hoping this was asked. Solution threw me off plus I don't understand how we account for Statement 2, the Prime component of it all.
gmat-admin's picture

Statement 2: The product of the smallest and largest integers in the list is a prime number.
The product of two prime numbers is composite (e.g., 3 x 5 = 15, and 15 is composite)
The product of a prime number and a composite number is composite (i.e., non-prime).
The product of two composite numbers is composite.

So, the are two ways for the product of two integers to be prime
- One number is 1 and the other number is prime (e.g., 1 x 7 = 7, and 7 is prime)
- One number is -1 and the other number is the negative of a prime number (e.g., -1 x -3 = 3, and 3 is prime)

I hope that helps.

Question: https://gmatclub.com/forum/if-each-of-the-two-digits-x-and-y-is-distinct-137998.html

Hi Brent,

For this question, I spent over 1 minute to confirm if 97 and 79 are primes. Do we need to memorize primes till 100 for the GMAT? Thanks.
gmat-admin's picture

Question link: https://gmatclub.com/forum/if-each-of-the-two-digits-x-and-y-is-distinct...

It's pretty rare for official GMAT questions to ask about primes greater than 70.
That said, even though this isn't an official question, there's certainly no harm in memorizing all of the primes up to 97.

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