For integers x, y, and z, if ((2^x)^y)^z = 131072 which of the following must be true ?
A. The product xyz is even
B. The product xyz is odd
C. The product xy is even
D. The product yz is prime
E. The product yz is positive
I can understand that the last digit will be odd (i.e. z = odd) but how to determine the rest, x & y ?
Tough PS - Number Properties & Exponents
This topic has expert replies
-
- Junior | Next Rank: 30 Posts
- Posts: 11
- Joined: Thu Oct 15, 2015 2:14 pm
- 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
As we know that no perfect square integer ends in either of the digits 2, 3, 7, or 8 and the result 131072 ends in 2; hence 131072 is not a perfect square. In other words, the exponent over 2 (i.e. xyz) is definitely not even, which means that [spoiler]xyz must be odd.late4thing wrote:For integers x, y, and z, if ((2^x)^y)^z = 131072 which of the following must be true ?
A. The product xyz is even
B. The product xyz is odd
C. The product xy is even
D. The product yz is prime
E. The product yz is positive
I can understand that the last digit will be odd (i.e. z = odd) but how to determine the rest, x & y ?
Pick B[/spoiler]
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
- MartyMurray
- Legendary Member
- Posts: 2131
- Joined: Mon Feb 03, 2014 9:26 am
- Location: https://martymurraycoaching.com/
- Thanked: 955 times
- Followed by:140 members
- GMAT Score:800
C, D and E can be eliminated because there is no way to determine these things about two out of the three exponents, because the third exponent can be positive, negative, or a fraction.late4thing wrote:For integers x, y, and z, if ((2^x)^y)^z = 131072 which of the following must be true ?
A. The product xyz is even
B. The product xyz is odd
C. The product xy is even
D. The product yz is prime
E. The product yz is positive
For instance, xy could be even or odd, because z could be anything. So if xyz were to have to be odd, and xy were even, then z could be a fraction having an odd number in the numerator and 2 in the denominator, making xyz odd.
So the choice becomes between A, the product xyz is even, and B, the product xyz is odd.
131072 ends in 2. So you can look for a pattern in the powers of 2.
2¹ = 2 ends in 2
2² = 4 ends in 4
2³ = 8 ends in 8
2� = 16 ends in 6
2� = 32 ends in 2
That a power of 2 ending in 2 is an odd power of 2 becomes clear.
The correct answer is B.
Alternate Method
C, D and E can be eliminated because there is no way to determine these things about two out of the three exponents, because the third exponent can be positive, negative, or a fraction.
If xyz is even or odd, then 131,072 must be an integer power of 2.
2¹� = 1024
So 131,072 = 1024 x (some integer power of 2 around 100).
The only integer power of 2 that is close to 100 is 2� = 128
So 131,072 = 1024 x 128 = 2¹� x 2�
10 + 7 = 17
The correct answer is B.
Marty Murray
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.
Perfect Scoring Tutor With Over a Decade of Experience
MartyMurrayCoaching.com
Contact me at [email protected] for a free consultation.
-
- GMAT Instructor
- Posts: 2630
- Joined: Wed Sep 12, 2012 3:32 pm
- Location: East Bay all the way
- Thanked: 625 times
- Followed by:119 members
- GMAT Score:780
If we have
2ˣʸᶻ = 131072 = 2¹�
we know that x * y * z = 17.
Beyond that, we can't find x, y, and z individually. We could have x = 17, y = 1, z = 1, or x = 17/2, y = 2, z = 1, etc. etc.
2ˣʸᶻ = 131072 = 2¹�
we know that x * y * z = 17.
Beyond that, we can't find x, y, and z individually. We could have x = 17, y = 1, z = 1, or x = 17/2, y = 2, z = 1, etc. etc.
- 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
Be careful friends and reread "For integers x, y, and z" to avoid consider fractional possibilities.
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
- Brent@GMATPrepNow
- GMAT Instructor
- Posts: 16207
- Joined: Mon Dec 08, 2008 6:26 pm
- Location: Vancouver, BC
- Thanked: 5254 times
- Followed by:1268 members
- GMAT Score:770
Related Resources
The following free videos provide information that's useful for answering this question:
Brent
The following free videos provide information that's useful for answering this question:
- - Laws of exponents - part I: https://www.gmatprepnow.com/module/gmat ... video/1025
- Units digits of large powers: https://www.gmatprepnow.com/module/gmat ... video/1031
Brent