Author 
Message 
TAGS:

Hide Tags

eGMAT Representative
Joined: 04 Jan 2015
Posts: 2000

If x and y are distinct positive integers. . .
[#permalink]
Show Tags
11 Nov 2016, 05:50
Question Stats:
51% (01:47) correct 49% (01:41) wrong based on 134 sessions
HideShow timer Statistics
If \(x\) and \(y\) are distinct positive integers and \(x+y\) is even, what is the remainder when \((x+y)^a\) is divided by \(10\), where \(a\) is a positive integer? (1) Units digit of \(y\) is \(6\) (2) \((xy)^a\) is divisible by \(10\).
Take a stab at this fresh question from eGMAT. Post your analysis below. Official Solution to be provided after receiving some good analyses.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Register for free sessions Number Properties  Algebra Quant Workshop
Success Stories Guillermo's Success Story  Carrie's Success Story
Ace GMAT quant Articles and Question to reach Q51  Question of the week
Must Read Articles Number Properties – Even Odd  LCM GCD  Statistics1  Statistics2 Word Problems – Percentage 1  Percentage 2  Time and Work 1  Time and Work 2  Time, Speed and Distance 1  Time, Speed and Distance 2 Advanced Topics Permutation and Combination 1  Permutation and Combination 2  Permutation and Combination 3  Probability Geometry Triangles 1  Triangles 2  Triangles 3  Common Mistakes in Geometry Algebra Wavy line  Inequalities Practice Questions Number Properties 1  Number Properties 2  Algebra 1  Geometry  Prime Numbers  Absolute value equations  Sets
 '4 out of Top 5' Instructors on gmatclub  70 point improvement guarantee  www.egmat.com



Current Student
Joined: 31 May 2015
Posts: 21

If x and y are distinct positive integers. . .
[#permalink]
Show Tags
11 Nov 2016, 08:46
I will try:
So basically, divisibility by 10 will require us to check the last digit so need to know last digit of x+y and how a is divisible by the corresponding cycle number. (1) is not enough because we know nothing about xlast digitinsufficient (2) (x.y)^a divisible by 10 so we have either x and y multiple of 2 and 5 or 10 and whatever the number Because x+y even so we get rid of 2 and 5 and go with 10 and whatever even number. However, we don't know the last digit of that whatever numberinsufficient
(1)+(2) so y must have last digit 6 and x must have last digit 0 > (x+y) last digit 6^a we know that cycle of 6 doesn't matter because the last digit is always 6, no matter a is> sufficient (C)



Current Student
Joined: 26 Jan 2016
Posts: 109
Location: United States
GPA: 3.37

Re: If x and y are distinct positive integers. . .
[#permalink]
Show Tags
11 Nov 2016, 12:58
1. Units digit of y is 6
from here all we know is that y is even and that x is even too x+y=even (given in prompt). There are a lot of different options for this. Insuff
2. (xy)^a/10 so from here we know that xy needs to be a multiple of 10. There are so many ways to do that. Insuff.
1&2. We know that the units digit of y is 6 so then X will have to involve a 0 as x+y=even. lets try 6 and 10. 6+10=16
16²=256/10=25 r =6
now lets try 6 and 20. 6+20=26 26²=676/10=67 r=6
C



eGMAT Representative
Joined: 04 Jan 2015
Posts: 2000

If x and y are distinct positive integers. . .
[#permalink]
Show Tags
Updated on: 14 Dec 2016, 20:54
Let's look at the detailed solution of the above problemSteps 1 & 2: Understand Question and Draw Inferences• x, y are distinct integers > 0 such that x + y = even • Hence we can have two possibilities
o both x and y are even OR o both x and y are odd • a is an integer > 0 To Find:The value of r in \((x+y)^a=10k+r\), where k is the quotient obtained when \((x+y)^a\) is divided by 10 and r is the remainder; so, \(0 ≤ r < 10\) o Now, when a number is divided by 10, the remainder is equal to the units digit of that number. o So, r = units digit of \((x+y)^a\)
Step 3: Analyze Statement 1 independentlyUnits digit of y is 6 • It does not tell us anything about the units digit of x as well as about the value of a.
So statement 1 is not sufficient to arrive at a unique answer. Step 4: Analyze Statement 2 independently \((xy)^a\) is divisible by 10. • As \((xy)^a\) is divisible by 10, the units digit of \((xy)^a\) = 0 • So, the units digit of xy = 0. Two cases are possible:
o Units digit of (x, y) = { 5, even number) in any order. However in this case the number with 5 as its units digit will be odd and the other number will be even. However, we’ve deduced in Steps 1 and 2 that x and y have the same evenodd nature. So, this case is not possible as it contradicts the given information (that the sum x + y is even). o Units digit of (x, y) = (0, even number) in any order. In this case x and y are both even. So, this case is possible. However since we do not have a unique value of units digit of both x and y and we do not know the value of a, we cannot find a unique value of the units digit of \((x+y)^a\) Therefore, statement 2 is sufficient to arrive at a unique answer.Step 5: Analyze Both Statements Together (if needed)1. From Statement 1, we know that Units digit of y = 6 2. From Statement 2, we inferred that Units digit of (x, y) = (0, even number) in any order Combining both the statements, we can say that units digit (x) = 0 and units digit(y) = 6 So, units digit of (x+y) = 6. Now do we need the value of a to find out the units digit of \((x+y)^a\)?We know a number with units digit of 6 raised to any power always results in units digit of 6. So, Units Digit of \(6^a = 6.\) Thus r = Units Digit of \(6^a = 6\).Sufficient to answer. Hence the correct Answer is CThanks, Saquib Quant Expert eGMAT
_________________
Register for free sessions Number Properties  Algebra Quant Workshop
Success Stories Guillermo's Success Story  Carrie's Success Story
Ace GMAT quant Articles and Question to reach Q51  Question of the week
Must Read Articles Number Properties – Even Odd  LCM GCD  Statistics1  Statistics2 Word Problems – Percentage 1  Percentage 2  Time and Work 1  Time and Work 2  Time, Speed and Distance 1  Time, Speed and Distance 2 Advanced Topics Permutation and Combination 1  Permutation and Combination 2  Permutation and Combination 3  Probability Geometry Triangles 1  Triangles 2  Triangles 3  Common Mistakes in Geometry Algebra Wavy line  Inequalities Practice Questions Number Properties 1  Number Properties 2  Algebra 1  Geometry  Prime Numbers  Absolute value equations  Sets
 '4 out of Top 5' Instructors on gmatclub  70 point improvement guarantee  www.egmat.com



eGMAT Representative
Joined: 04 Jan 2015
Posts: 2000

Re: If x and y are distinct positive integers. . .
[#permalink]
Show Tags
14 Dec 2016, 07:41
hanminhee wrote: I will try:
So basically, divisibility by 10 will require us to check the last digit so need to know last digit of x+y and how a is divisible by the corresponding cycle number. (1) is not enough because we know nothing about xlast digitinsufficient (2) (x.y)^a divisible by 10 so we have either x and y multiple of 2 and 5 or 10 and whatever the number Because x+y even so we get rid of 2 and 5 and go with 10 and whatever even number. However, we don't know the last digit of that whatever numberinsufficient
(1)+(2) so y must have last digit 6 and x must have last digit 0 > (x+y) last digit 6^a we know that cycle of 6 doesn't matter because the last digit is always 6, no matter a is> sufficient (C) Hey Hanminhee, The analysis presented by you is absolutely correct! You have solved the question in a very methodical way, and that is how we encourage students to solve any question at eGMAT. Thanks Saquib Quant Expert eGMAT
_________________
Register for free sessions Number Properties  Algebra Quant Workshop
Success Stories Guillermo's Success Story  Carrie's Success Story
Ace GMAT quant Articles and Question to reach Q51  Question of the week
Must Read Articles Number Properties – Even Odd  LCM GCD  Statistics1  Statistics2 Word Problems – Percentage 1  Percentage 2  Time and Work 1  Time and Work 2  Time, Speed and Distance 1  Time, Speed and Distance 2 Advanced Topics Permutation and Combination 1  Permutation and Combination 2  Permutation and Combination 3  Probability Geometry Triangles 1  Triangles 2  Triangles 3  Common Mistakes in Geometry Algebra Wavy line  Inequalities Practice Questions Number Properties 1  Number Properties 2  Algebra 1  Geometry  Prime Numbers  Absolute value equations  Sets
 '4 out of Top 5' Instructors on gmatclub  70 point improvement guarantee  www.egmat.com



Manager
Status: One Last Shot !!!
Joined: 04 May 2014
Posts: 241
Location: India
Concentration: Marketing, Social Entrepreneurship
GMAT 1: 630 Q44 V32 GMAT 2: 680 Q47 V35

Re: If x and y are distinct positive integers. . .
[#permalink]
Show Tags
14 Dec 2016, 19:17
EgmatQuantExpert wrote: So, units digit of (x+y) = 6. Now do we need the value of a to find out the units digit of (x+y)?
A small typo in the above line: Now do we need the value of a to find out the units digit of \((x+y)^a\)?Thanks for the detailed explanation!
_________________
One Kudos for an everlasting piece of knowledge is not a bad deal at all...
 Twenty years from now you will be more disappointed by the things you didn't do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover. Mark Twain



Senior Manager
Joined: 05 Dec 2016
Posts: 251
Concentration: Strategy, Finance

If x and y are distinct positive integers. . .
[#permalink]
Show Tags
16 Feb 2017, 04:32
(1) Implies that X is even. Not Suff. (2) Implies that XY should be a multiple of 10, and have at least 2 & 5 in its base. Different options are possible. Not Suff. (1) + (2) Since X is even, it cannot be 5, so XY to be a multiple of 10, X must be equal to 10 at least. That gives us understanding that any sum of X and Y will yield in 6 as units digit, Suff. Answer C.




If x and y are distinct positive integers. . . &nbs
[#permalink]
16 Feb 2017, 04:32






