If k is an integer, and (35^2 - 1)/k is an integer, then k could be each of the following, EXCEPT
(A) 8
(B) 9
(C) 12
(D) 16
(E) 17
[spoiler]Source: https://readyforgmat.com[/spoiler]
5-Day Free Trial
5-day free, full-access trial TTP
Available with Beat the GMAT members only code
MORE DETAILS
(35^2 – 1)/k is an integer
This topic has expert replies
-
sanju09
- GMAT Instructor
- Posts: 3650
- Joined: Wed Jan 21, 2009 4:27 am
- Location: India
- Thanked: 267 times
- Followed by:80 members
- GMAT Score:760
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
GMAT/MBA Expert
-
Rahul@gurome
- GMAT Instructor
- Posts: 1179
- Joined: Sun Apr 11, 2010 9:07 pm
- Location: Milpitas, CA
- Thanked: 447 times
- Followed by:88 members
(35^2 - 1)/k = (35 - 1)(35 + 1)/k = (34)(36)/k
We can clearly see that if k = 17, 12, 9 or 8, then (35^2 - 1)/k will give an integer value.
But if k = 16, then 35^2 - 1)/k will not give an integer value.
The correct answer is [spoiler](D)[/spoiler].
We can clearly see that if k = 17, 12, 9 or 8, then (35^2 - 1)/k will give an integer value.
But if k = 16, then 35^2 - 1)/k will not give an integer value.
The correct answer is [spoiler](D)[/spoiler].
Rahul Lakhani
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
Quant Expert
Gurome, Inc.
https://www.GuroMe.com
On MBA sabbatical (at ISB) for 2011-12 - will stay active as time permits
1-800-566-4043 (USA)
+91-99201 32411 (India)
-
Brian@VeritasPrep
- GMAT Instructor
- Posts: 1031
- Joined: Thu Jul 03, 2008 1:23 pm
- Location: Malibu, CA
- Thanked: 716 times
- Followed by:255 members
- GMAT Score:750
Hey everyone,
Rahul's solution above is terrific, but let me point out a few strategic things here.
1) When you see that (35^2 - 1) setup, or any version of (x^2 - y^2), you HAVE TO think about the Difference of Squares rule: x^2 - y^2 = (x+y)(x-y). That rule sets up quite a bit for you - namely, it allows you to turn ugly subtraction into multiplication, which should allow you to factor.
2) Pursuant to the above, when you see addition/subtraction of exponents, there's an overwhelming likelihood that you'll need to factor somehow to turn that into multiplication (broken record for those of you reading my posts today, but I can't stress this enough). Difference of Squares is a great tool to have in your arsenal for that.
3) You may find it helpful to break down the resulting divisibility problem (34*36/k) into prime factors:
2*2*2*3*3*17 is divisible by k.
Since the answer choices are:
8 = 2*2*2 (divisible)
9 = 3*3 (divisible)
12 = 2*2*3 (divisible)
16 = 2*2*2*2 (we're one 2 short, so this is NOT divisible)
17 = 17 (divisible)
You can attack these systematically. When divisibility is in question, prime factorization helps you to have a system for checking divisibility without needing to treat each division as individual.
Rahul's solution above is terrific, but let me point out a few strategic things here.
1) When you see that (35^2 - 1) setup, or any version of (x^2 - y^2), you HAVE TO think about the Difference of Squares rule: x^2 - y^2 = (x+y)(x-y). That rule sets up quite a bit for you - namely, it allows you to turn ugly subtraction into multiplication, which should allow you to factor.
2) Pursuant to the above, when you see addition/subtraction of exponents, there's an overwhelming likelihood that you'll need to factor somehow to turn that into multiplication (broken record for those of you reading my posts today, but I can't stress this enough). Difference of Squares is a great tool to have in your arsenal for that.
3) You may find it helpful to break down the resulting divisibility problem (34*36/k) into prime factors:
2*2*2*3*3*17 is divisible by k.
Since the answer choices are:
8 = 2*2*2 (divisible)
9 = 3*3 (divisible)
12 = 2*2*3 (divisible)
16 = 2*2*2*2 (we're one 2 short, so this is NOT divisible)
17 = 17 (divisible)
You can attack these systematically. When divisibility is in question, prime factorization helps you to have a system for checking divisibility without needing to treat each division as individual.
Last edited by Brian@VeritasPrep on Tue Sep 14, 2010 11:59 am, edited 1 time in total.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
-
ankur.agrawal
- Master | Next Rank: 500 Posts
- Posts: 261
- Joined: Wed Mar 31, 2010 8:37 pm
- Location: Varanasi
- Thanked: 11 times
- Followed by:3 members
Thx Brian not only for solving the problems but also for giving us deeper insights.Brian@VeritasPrep wrote:Hey everyone,
Rahul's solution above is terrific, but let me point out a few strategic things here.
1) When you see that (35^2 - 1) setup, or any version of (x^2 - y^2), you HAVE TO think about the Difference of Squares rule: x^2 - y^2 = (x+y)(x-y). That rule sets up quite a bit for you - namely, it allows you to turn ugly subtraction into multiplication, which should allow you to factor.
2) Pursuant to the above, when you see addition/subtraction of exponents, there's an overwhelming likelihood that you'll need to factor somehow to turn that into multiplication (broken record for those of you reading my posts today, but I can't stress this enough). Difference of Squares is a great tool to have in your arsenal for that.
3) You may find it helpful to break down the resulting divisibility problem (34*36/k) into prime factors:
2*2*2*3*3*17 is divisible by k.
Since the answer choices are:
8 = 2*2*2 (divisible)
9 = 3*3*3 (divisible)
12 = 2*2*3 (divisible)
16 = 2*2*2*2 (we're one 2 short, so this is NOT divisible)
17 = 17 (divisible)
You can attack these systematically. When divisibility is in question, prime factorization helps you to have a system for checking divisibility without needing to treat each division as individual.
-
ankur.agrawal
- Master | Next Rank: 500 Posts
- Posts: 261
- Joined: Wed Mar 31, 2010 8:37 pm
- Location: Varanasi
- Thanked: 11 times
- Followed by:3 members
Brian@VeritasPrep wrote:Hey everyone,
Rahul's solution above is terrific, but let me point out a few strategic things here.
1) When you see that (35^2 - 1) setup, or any version of (x^2 - y^2), you HAVE TO think about the Difference of Squares rule: x^2 - y^2 = (x+y)(x-y). That rule sets up quite a bit for you - namely, it allows you to turn ugly subtraction into multiplication, which should allow you to factor.
2) Pursuant to the above, when you see addition/subtraction of exponents, there's an overwhelming likelihood that you'll need to factor somehow to turn that into multiplication (broken record for those of you reading my posts today, but I can't stress this enough). Difference of Squares is a great tool to have in your arsenal for that.
3) You may find it helpful to break down the resulting divisibility problem (34*36/k) into prime factors:
2*2*2*3*3*17 is divisible by k.
Since the answer choices are:
8 = 2*2*2 (divisible)
9 = 3*3*3 (divisible)
12 = 2*2*3 (divisible)
16 = 2*2*2*2 (we're one 2 short, so this is NOT divisible)
17 = 17 (divisible)
You can attack these systematically. When divisibility is in question, prime factorization helps you to have a system for checking divisibility without needing to treat each division as individual.
how did u write 9= 3*3*3 above.
-
Brian@VeritasPrep
- GMAT Instructor
- Posts: 1031
- Joined: Thu Jul 03, 2008 1:23 pm
- Location: Malibu, CA
- Thanked: 716 times
- Followed by:255 members
- GMAT Score:750
ankur.agrawal wrote:Typo...the 3 key must have stuck! Sorry about that...just fixed it.Brian@VeritasPrep wrote:Hey everyone,
Rahul's solution above is terrific, but let me point out a few strategic things here.
1) When you see that (35^2 - 1) setup, or any version of (x^2 - y^2), you HAVE TO think about the Difference of Squares rule: x^2 - y^2 = (x+y)(x-y). That rule sets up quite a bit for you - namely, it allows you to turn ugly subtraction into multiplication, which should allow you to factor.
2) Pursuant to the above, when you see addition/subtraction of exponents, there's an overwhelming likelihood that you'll need to factor somehow to turn that into multiplication (broken record for those of you reading my posts today, but I can't stress this enough). Difference of Squares is a great tool to have in your arsenal for that.
3) You may find it helpful to break down the resulting divisibility problem (34*36/k) into prime factors:
2*2*2*3*3*17 is divisible by k.
Since the answer choices are:
8 = 2*2*2 (divisible)
9 = 3*3*3 (divisible)
12 = 2*2*3 (divisible)
16 = 2*2*2*2 (we're one 2 short, so this is NOT divisible)
17 = 17 (divisible)
You can attack these systematically. When divisibility is in question, prime factorization helps you to have a system for checking divisibility without needing to treat each division as individual.
how did u write 9= 3*3*3 above.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
GMAT Instructor
Chief Academic Officer
Veritas Prep
Looking for GMAT practice questions? Try out the Veritas Prep Question Bank. Learn More.
-
sanju09
- GMAT Instructor
- Posts: 3650
- Joined: Wed Jan 21, 2009 4:27 am
- Location: India
- Thanked: 267 times
- Followed by:80 members
- GMAT Score:760
very laboriousklmehta03 wrote:{[(35^2) -1]/k}=integer
integer= (35-1)*(35+1)/k
=1224/k
only when you divide by 16 the divisor is not an integer
so ans D
The mind is everything. What you think you become. -Lord Buddha
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
Sanjeev K Saxena
Quantitative Instructor
The Princeton Review - Manya Abroad
Lucknow-226001
www.manyagroup.com
-
anirudhbhalotia
- Master | Next Rank: 500 Posts
- Posts: 123
- Joined: Tue Nov 23, 2010 7:18 pm
- Location: Mumbai, India
- Thanked: 5 times
- Followed by:4 members
Brian@VeritasPrep wrote:Hey everyone,
Rahul's solution above is terrific, but let me point out a few strategic things here.
1) When you see that (35^2 - 1) setup, or any version of (x^2 - y^2), you HAVE TO think about the Difference of Squares rule: x^2 - y^2 = (x+y)(x-y). That rule sets up quite a bit for you - namely, it allows you to turn ugly subtraction into multiplication, which should allow you to factor.
2) Pursuant to the above, when you see addition/subtraction of exponents, there's an overwhelming likelihood that you'll need to factor somehow to turn that into multiplication (broken record for those of you reading my posts today, but I can't stress this enough). Difference of Squares is a great tool to have in your arsenal for that.
3) You may find it helpful to break down the resulting divisibility problem (34*36/k) into prime factors:
2*2*2*3*3*17 is divisible by k.
Since the answer choices are:
8 = 2*2*2 (divisible)
9 = 3*3 (divisible)
12 = 2*2*3 (divisible)
16 = 2*2*2*2 (we're one 2 short, so this is NOT divisible)
17 = 17 (divisible)
You can attack these systematically. When divisibility is in question, prime factorization helps you to have a system for checking divisibility without needing to treat each division as individual.
Nice! Thanks Brian!
You mentioned Difference of Square Rule, if there is a "+" sign, can we still break it up as (x^2 + y^2)?
In the OG-12 Maths review is this rule mentioned...I don't remember it seeing in the book nor in the Gmat Prep!
Can you share/mention more of such essential concepts which we will definitely need to apply? This will be a HUGE help!
Thanks!
-
fskilnik@GMATH
- GMAT Instructor
- Posts: 1449
- Joined: Sat Oct 09, 2010 2:16 pm
- Thanked: 59 times
- Followed by:33 members
Another (more theoretical) approach: (It is what is "behind" Brian´s (3) item, watch out!)
When you are told that a and b are integers such that a/b is an integer (b implicitly non-zero, for sure), that means that b is a divisor of a.
Therefore, from the question stem we know that k must be a divisor of 35^2-1 = (34)(36) = 2^3*3^2*17.
(A) 2^3 is a divisor of the "red thing", for sure.
(B) 3^2 also
(C) 2^2*3 also
(D) 2^4 is NOT, because we have only three 2´s and we "would need" four 2´s.
PUFF!
Regards,
Fabio.
When you are told that a and b are integers such that a/b is an integer (b implicitly non-zero, for sure), that means that b is a divisor of a.
Therefore, from the question stem we know that k must be a divisor of 35^2-1 = (34)(36) = 2^3*3^2*17.
(A) 2^3 is a divisor of the "red thing", for sure.
(B) 3^2 also
(C) 2^2*3 also
(D) 2^4 is NOT, because we have only three 2´s and we "would need" four 2´s.
PUFF!
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
i don't think that we can "clearly" see that if k = 17, 12, 9, or 8, then (35^2-1)/k will give an integer value.
for (34)(36)/k, i think we can clearly see that 17, 12, and 9 all result in integer values (they are factors of either 34 or 36). However, neither 8 nor 16 are factors of 34 or 36, and the math required to multiply 34 by 36 then divide out 8 and 16 would be too time consuming (we could factor a 2 out, leaving 17*72, but that isn't necessarily clear on first glance either).
What we are left with is both options A (8) and D (16). Because 8 is a factor of 16, if the number is divisible by 16, then it must also be divisible by 8, however, the converse is not true (a number divisible by 8 is not necessarily divisible by 16). Therefore, 16 must be the number which results in a non-integer result.
for (34)(36)/k, i think we can clearly see that 17, 12, and 9 all result in integer values (they are factors of either 34 or 36). However, neither 8 nor 16 are factors of 34 or 36, and the math required to multiply 34 by 36 then divide out 8 and 16 would be too time consuming (we could factor a 2 out, leaving 17*72, but that isn't necessarily clear on first glance either).
What we are left with is both options A (8) and D (16). Because 8 is a factor of 16, if the number is divisible by 16, then it must also be divisible by 8, however, the converse is not true (a number divisible by 8 is not necessarily divisible by 16). Therefore, 16 must be the number which results in a non-integer result.
Rahul@gurome wrote:(35^2 - 1)/k = (35 - 1)(35 + 1)/k = (34)(36)/k
We can clearly see that if k = 17, 12, 9 or 8, then (35^2 - 1)/k will give an integer value.
But if k = 16, then 35^2 - 1)/k will not give an integer value.
The correct answer is [spoiler](D)[/spoiler].
-
Vinoth9132*
- Newbie | Next Rank: 10 Posts
- Posts: 1
- Joined: Wed Sep 18, 2013 1:34 am
