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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION.

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stonecold
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   03 Mar 2018, 00:56
loserunderachiever wrote:
Stone ,

Can you give me a solution of these two ?



Sure!
Let me try ->
Question 1)
Here, just for the ease of calculation -> let us assume that \(25 = a\) and \(10√6= b\) such that \(T = √a+b + √a-b\)

\(T = √a+b + √a-b\)
\(T^2 = a+b + a - b + 2√a+b * √a-b = 2a + 2√a^2-b^2\)

Now, put back the values of a and b
\(T^2 = 50 + 2√625-600\)
\(T^2 = 50 + 2*√25\)
\(T^2 = 50 + 10 = 60\)
\(T = √60\)
\(T = 2√15\)

SMASH that C!


Question 2) Now, This one is tricky. But notice that this is an infinite loop. So there must be a major trick to solve this.
We can actually replace the value in the root symbol after the first 6 with x.
\(x = √(6+x)\)
\(x^2 = 6+x\)
\(x^2-x-6 = 0\)
\(x^2-3x+2x-6=0\)
\(x(x-3)+2(x-3)=0\)
\((x-3)(x+2)=0\)
\(x=3\) or \(x=-2\)

But x cannot be negative since its a root value and a root is always passive. So x≠-2

Hence x = 3
SMASH that B!
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   03 Mar 2018, 02:13
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loserunderachiever wrote:
Stone ,

Can you give me a solution of these two ?


First question is discussed here: https://gmatclub.com/forum/new-tough-an ... l#p1029216

Second question is discussed here: https://gmatclub.com/forum/new-tough-an ... l#p1029228

PLEASE FOLLOW THE RULES WHEN POSTING QUESTIONS: https://gmatclub.com/forum/rules-for-po ... 33935.html Questions should NOT be posted in this thread they should be posted in respective forums! Thank you.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   07 Mar 2018, 23:27
Hi stonecold

just a quick ask... this question is from Mock 1 and I am just wondering if the answer is correct. Please let me know as I am getting 270 total factors.
2^8*3^4*5^2*7^1

so basically = (8+1)(4+1)(2+1)(1+1)

48)If n is the product of all the integers from 1 to 10 exclusive,how many factors does n have?
[Obscure] Spoiler:
160.
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   07 Mar 2018, 23:39
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Bobzi wrote:
Hi stonecold

just a quick ask... this question is from Mock 1 and I am just wondering if the answer is correct. Please let me know as I am getting 270 total factors.
2^8*3^4*5^2*7^1

so basically = (8+1)(4+1)(2+1)(1+1)

48)If n is the product of all the integers from 1 to 10 exclusive,how many factors does n have?
[Obscure] Spoiler:
160.



Hi,
Your solution is incorrect.

See ->

\(2*3*4*5*6*7*8*9= 2^7*3^4*5*7\)

=> # of factors = \(8*5*2*2 = 160\)

Your mistake -> You did not read the question properly. Notice the word EXCLUSIVE. You must EXCLUDE 1 and 10.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   09 Mar 2018, 06:07
This is a gold mine of information!
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   11 Apr 2018, 11:51
DebUSA wrote:
This is a gold mine of information!


I am glad you found it useful. I will be updating the tests soon.

All the best :cool: :thumbup:
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   25 Apr 2018, 08:12
14)If p is a positive integer and 200 multiplies by p is square of an integer,what is the value of p?
A)2
B)3
C)5
D)10
E)Cannot be determined.

HI stonecold,

\(x^2 = p*200\) In this question why can't p =2 ?
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   26 May 2018, 22:57
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Firstly, thanks for having these questions - they are a great help!

Wanted to ask, though, about the question below:

7)What is the units digit of 23^99∗14^352+9002^1003∗918^437986

I keep getting 7 * 1 + 8 * 4 = 9. I'm using cyclicity here. Thanks in advanced.
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   27 May 2018, 00:59
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wchin24 wrote:
Firstly, thanks for having these questions - they are a great help!

Wanted to ask, though, about the question below:

7)What is the units digit of 23^99∗14^352+9002^1003∗918^437986

I keep getting 7 * 1 + 8 * 4 = 9. I'm using cyclicity here. Thanks in advanced.


Hey wchin24 ,

The mistake you did is highlighted above.

14^352

The pattern of 4 is 4,6

This means cyclicity is 2.

When you divide the power with 2, you will get 0 remainder. That means last digit is last place of the pattern, which is 6 here.

Hence, the last digit of 14^352 is 6.

Does that make sense?
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   27 May 2018, 01:16
abhimahna wrote:
wchin24 wrote:
Firstly, thanks for having these questions - they are a great help!

Wanted to ask, though, about the question below:

7)What is the units digit of 23^99∗14^352+9002^1003∗918^437986

I keep getting 7 * 1 + 8 * 4 = 9. I'm using cyclicity here. Thanks in advanced.


Hey wchin24 ,

The mistake you did is highlighted above.

14^352

The pattern of 4 is 4,6

This means cyclicity is 2.

When you divide the power with 2, you will get 0 remainder. That means last digit is last place of the pattern, which is 6 here.

Hence, the last digit of 14^352 is 6.

Does that make sense?


OMG YES! Thank you! I was tripping up since I was seeing it as 4^0, but in reality this just means that it'll end up at the last place of the pattern here.
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   27 May 2018, 20:31
NandishSS wrote:
14)If p is a positive integer and 200 multiplies by p is square of an integer,what is the value of p?
A)2
B)3
C)5
D)10
E)Cannot be determined.

HI stonecold,

\(x^2 = p*200\) In this question why can't p =2 ?


Hi, You are right in saying that p CAN be 2. But you missing the big picture here.
Ask yourself, is that the only possible value of p?
What if p is \(2*5^2\) or \(2*11^2\) or \(2*101^2\) ?

You see, there are various value of the variable p that are possible in thus question. Hence the OA is E.
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   02 Aug 2018, 20:30
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121)Data Sufficiency->How many divisors does the positive integer N have?

(1)27N^3 has 16 factors.
(2)90<N^3<200

as explained, how can we say that N is prime from Statement 1. N could be 3^4 as well. If N= 3^4 then 27N^3 = 3^3*((3^4)^3)= 3^3*3^12= 3^15 Hence Factors will be 15+1=16. And Nos of Divisor of N = 3^4 is Five. Not 2 as in the case if N is prime.

Please explain if anything wrong in my understanding
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   28 Jan 2019, 02:38
Hi stonecold

For question 18 It looks like answer should be C and not E as only possible option after combining both the statement is P = 3, Could you please advise if I am missing something.

chetan2u, Bunuel, VeritasKarishma, gmatbusters, amanvermagmat

stonecold wrote:
18)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is a factor of 75

Spoiler: ::
E.
Combing the two statements -> p can be 3 or 3*5^2
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   28 Jan 2019, 02:42
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Gmatprep550 wrote:
Hi stonecold

For question 18 It looks like answer should be C and not E as only possible option after combining both the statement is P = 3, Could you please advise if I am missing something.

chetan2u, Bunuel, VeritasKarishma, gmatbusters, amanvermagmat

stonecold wrote:
18)Data Sufficiency->What is the value of positive integer p?
A)300 multiplied by p is square of an integer.
B)p is a factor of 75

Spoiler: ::
E.
Combing the two statements -> p can be 3 or 3*5^2



No, both 3 and 75 will fit in..
A)300 multiplied by p is square of an integer...300*3=900=30^2 and 300*75=22500=150^2
B)p is a factor of 75... Factors of 75 are 1,3,5,15,25,75, so both 3 and 75 fit in
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   28 Jan 2019, 04:27
Hi chetan2u, Bunuel, VeritasKarishma, Gladiator59, generis

For question 23 statement 7-8

I am only able to find that "All the prime numbers greater than 3 can be written as either 6n+1 or 6n-1." Not able to find anything for 6+1 and 4n+1 or 4n-1.

Could you please review and advise if I am missing something.

stonecold wrote:
23)Which of the following statements must be true->
1)A prime number must be positive.
2)For any prime number p,there is no x such that 1<x<p and x is a divisor of p.
3)The product of first ten primes is even.
4)All prime numbers greater than 71 are odd.
5)2 and 3 are the only consecutive integers that are also prime numbers.
6)p is a prime number and x and y are positive integers.If p=x*y then one out of x or y must be one.
7)All the prime numbers greater than 3 can be written as either 4n+1 or 4n-1.
8)All the prime numbers greater than 3 can be written as either 6+1 or 6n-1.

Spoiler: ::
All statements are true.
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   28 Jan 2019, 05:09
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Gmatprep550 wrote:
Hi chetan2u, Bunuel, VeritasKarishma, Gladiator59, generis

For question 23 statement 7-8

I am only able to find that "All the prime numbers greater than 3 can be written as either 6n+1 or 6n-1." Not able to find anything for 6+1 and 4n+1 or 4n-1.

Could you please review and advise if I am missing something.

stonecold wrote:
23)Which of the following statements must be true->
1)A prime number must be positive.
2)For any prime number p,there is no x such that 1<x<p and x is a divisor of p.
3)The product of first ten primes is even.
4)All prime numbers greater than 71 are odd.
5)2 and 3 are the only consecutive integers that are also prime numbers.
6)p is a prime number and x and y are positive integers.If p=x*y then one out of x or y must be one.
7)All the prime numbers greater than 3 can be written as either 4n+1 or 4n-1.
8)All the prime numbers greater than 3 can be written as either 6+1 or 6n-1.

Spoiler: ::
All statements are true.


Hi..

All prime numbers have to be of type of 6n+1 or 6n-1

Now 4n+1 and 4n-1 is nothing but set of all odd numbers..
When n=1, 4n+1 and 4n-1 becomes 3 and 5 and when n=2, 4n+1 and 4n-1 becomes 7 and 9..so 3,5,7,9,11,...
And all primes above 2 are odd numbers..so primes are also of type 4n+1 or 4n-1, basically they will be odd
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   28 Jan 2019, 05:43
Thanks chetan2u for valuable response. It helped me :)
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STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   28 Jan 2019, 09:24
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As perfectly explained by chetan2u Sir,

All prime numbers greater than 3 can be written as 6n+/-1 or 4n+/-1.

But vice versa is not true
for example, for n =4, 6*4+1= 25 , but it is not prime
Similarly, for n = 2, 4n+1 = 2*4+1= 9, which is not prime


In fact, there is no direct formula, which can find whether a number is prime or not.

All prime numbers greater than 3 can be written as 6n+/-1 or 4n+/-1, but all numbers in form of 6n+/-1 or 4n+/-1 are not prime.



Gmatprep550 wrote:
Hi chetan2u, Bunuel, VeritasKarishma, Gladiator59, generis

For question 23 statement 7-8

I am only able to find that "All the prime numbers greater than 3 can be written as either 6n+1 or 6n-1." Not able to find anything for 6+1 and 4n+1 or 4n-1.

Could you please review and advise if I am missing something.

stonecold wrote:
23)Which of the following statements must be true->
1)A prime number must be positive.
2)For any prime number p,there is no x such that 1<x<p and x is a divisor of p.
3)The product of first ten primes is even.
4)All prime numbers greater than 71 are odd.
5)2 and 3 are the only consecutive integers that are also prime numbers.
6)p is a prime number and x and y are positive integers.If p=x*y then one out of x or y must be one.
7)All the prime numbers greater than 3 can be written as either 4n+1 or 4n-1.
8)All the prime numbers greater than 3 can be written as either 6+1 or 6n-1.

Spoiler: ::
All statements are true.
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   18 Feb 2019, 14:18
@14)If p is a positive integer and 200 multiplies by p is square of an integer,what is the value of p?
A)2
B)3
C)5
D)10
E)Cannot be determined.


Hi, can you explain, how have you written p=2*2^2 expression in the first set of questions, numbered 15-17..
What does this expression mean? i am confused in this and the DS questions that contain 300 and 200 too
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink] New post   01 Mar 2020, 15:54
where is the link of mock test 1 and 2?
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Re: STONECOLD'S MATH CHALLENGE - PS AND DS QUESTION COLLECTION. [#permalink]
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