Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 49271

Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
14 Sep 2015, 23:04
Question Stats:
65% (02:16) correct 35% (02:38) wrong based on 106 sessions
HideShow timer Statistics
Starting with 0, a mathematician labels every nonnegative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power? A. alpha B. beta C. gamma D. delta E. epsilon Kudos for a correct solution.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



SVP
Joined: 08 Jul 2010
Posts: 2314
Location: India
GMAT: INSIGHT
WE: Education (Education)

Re: Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
14 Sep 2015, 23:52
Bunuel wrote: Starting with 0, a mathematician labels every nonnegative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power?
A. alpha B. beta C. gamma D. delta E. epsilon
Kudos for a correct solution. alpha  0, 5, 10, 15, 20, .... beta  1, 6, 11, 16, 21, ... gamma  2, 7, 12, 17, 22, ... delta  3, 8, 13, 18, 23, ... epsilon  4, 9, 14, 19, 24, ... An integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power  (Gamma)^7 + (Delta)^7 = 2^7 + 3^7 The cyclicity of the Labels is 5 i.e. after every 5 consecutive non negative numbers the labels repeat so, Remainder when 2^7 + 3^7 is divided by 5 = Remainder when 2^7 + (2)^7 is divided by 5 = R(128/5) + R(128/5) = +3  3 = 0 (which falls in label 'Alpha") i.e. First step of cycle  "Alpha"Answer: option A
_________________
Prosper!!! GMATinsight Bhoopendra Singh and Dr.Sushma Jha email: info@GMATinsight.com I Call us : +919999687183 / 9891333772 Online OneonOne Skype based classes and Classroom Coaching in South and West Delhi http://www.GMATinsight.com/testimonials.html
22 ONLINE FREE (FULL LENGTH) GMAT CAT (PRACTICE TESTS) LINK COLLECTION



Manager
Joined: 10 Aug 2015
Posts: 103

Re: Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
15 Sep 2015, 02:26
Bunuel wrote: Starting with 0, a mathematician labels every nonnegative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power?
A. alpha B. beta C. gamma D. delta E. epsilon
Kudos for a correct solution. Solution : 2 is gamma and 3 is a delta. 2^7 + 3^7 will have a units digit as 8+7=5. So, label is alpha. or alpha : 5k beta : 5k +1 gamma : 5K+2 delta : 5k+3 epsilon : 5K+4 (5k + 2)^7 has unit digit as 8 or 3 (5k + 3)^7 has unit digit as 7 or 2 In any case (5k + 2)^7 + (5k + 3)^7 will have units digit as 0 or 5 which is a alpha. Option A



Manager
Joined: 13 Mar 2013
Posts: 171
Location: United States
Concentration: Leadership, Technology
GPA: 3.5
WE: Engineering (Telecommunications)

Re: Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
15 Sep 2015, 20:17
Starting with 0, a mathematician labels every nonnegative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power? A. alpha B. beta C. gamma D. delta E. epsilon 0  A alpha 1B beta 2G gamma 3D delta 4E epsilon 5A 6B 7G 8D 9E 10A sum of a gamma raised to the seventh power and a delta raised to the seventh power ? lets my smallest gamma be 2 and delta be 3 So 2^7 + 3^7 = 128 +2187 = 2315 Now see the last digit in the sum is 5 . And now check the above list  We get 0  A alpha ,5 A alpha and 10  A alpha  which means all the multiple of 5 are alpha And in our sum (2315)we also got a number which is multiple of 5 . Hence A ans .. Kudos please if my solution is simple and easy to understand .
_________________
Regards ,



Manager
Joined: 29 Jul 2015
Posts: 159

Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
17 Sep 2015, 13:53
Bunuel wrote: Starting with 0, a mathematician labels every nonnegative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power?
A. alpha B. beta C. gamma D. delta E. epsilon
Kudos for a correct solution. AlphaBetaGammaDeltaEpsilon 01234 56789 1011121314 1516171819 From the table it is clear that each number belongs to different group based on it's unit digit. We need to find the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power. This can be done by knowing the unit's digit of this integer. Let's take 2 from gamma and 3 from delta. Then the unit's digit of \(2^7+3^7\) Both 2 and 3 follow a cycle of 4 for unit's digit. So, the unit's will be that of \(2^3+3^3\) i.e 5. From the table we can see that it belongs to Alpha. Answer: A



Math Expert
Joined: 02 Sep 2009
Posts: 49271

Re: Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
20 Sep 2015, 21:09
Bunuel wrote: Starting with 0, a mathematician labels every nonnegative integer as one of five types: alpha, beta, gamma, delta, or epsilon, in that repeating order as the integers increase. For instance, the integer 8 is labeled delta. What is the label on an integer that is the sum of a gamma raised to the seventh power and a delta raised to the seventh power?
A. alpha B. beta C. gamma D. delta E. epsilon
Kudos for a correct solution. MANHATTAN GMAT OFFICIAL SOLUTION:Since there are five labels, given in order to all the integers, the label alpha is given to 0, 5, 10, etc. – that is, the alpha’s are the multiples of 5 and end in 0 or 5. All the other labels correspond to nonmultiples of 5 – in fact, they each correspond to particular remainders and particular units digits. For instance, the beta’s (1, 6, 11, 16, etc.), which all end in 1 or 6, also all leave a remainder of 1 after division by 5. The gamma’s correspond to a remainder of 2 (units digits = 2 or 7). Delta’s correspond to a remainder of 3 (units digits = 3 or 8), and epsilon’s correspond to a remainder of 4 (units digits = 4 or 9). Now, a gamma raised to the seventh power will be large, even if we pick the smallest gamma (2 itself). But all we need is the units digit of the result. So compute the units digit in stages: First power: units digit = 2 Second power: units digit = 2×2 = 4 Third power: units digit = 2×4 = 8 (remainder = 3) Fourth power: units digit = 2×8 = 16 = …6 (units digit only) (remainder = 1) Fifth power: units digit = 2×6 = 12 = …2 (units digit only) (remainder = 2) Sixth power: units digit = 2×2 = 4 (remainder = 4) Seventh power: units digit = 2×4 = 8 (remainder = 3) Do the same for the delta. First power: units digit = 3 Second power: units digit = 3×3 = 9 (remainder = 4) Third power: units digit = 3×9 = 27 = …7 (remainder = 2) Fourth power: units digit = 3×7 = 21 = …1 (remainder = 1) Fifth power: units digit = 3×1 = 3 (remainder = 3) Sixth power: units digit = 3×3 = 9 (remainder = 4) Seventh power: units digit = 3×9 = 27 = …7 (remainder = 2) \(G a m m a^7\) gives us a remainder of 3. \(D e l t a^7\) gives us a remainder of 2. Adding the remainders, we get a remainder of 5, which is the same as a remainder of 0 (remember, we’re talking about division by 5). So the sum gets a label of alpha. The correct answer is A.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 17 May 2018
Posts: 32

Re: Starting with 0, a mathematician labels every nonnegative integer as
[#permalink]
Show Tags
03 Sep 2018, 07:04
To find the label to assign to a number, we need to find the last digit of that number. Let's take the first gamma, 2, and raise it to 7. The last digit will be a 8. Let's take the following gamma, 7, and raise it to 7. The last digit will be 3. Now let's take the first delta, 3, and raise it to 7. The last digit will be a 7. Let's take the following delta, 8, and raise it to 7. The last digit will be 2. The sum of any gamma raised to 7 and any delta raised to 7 will result in a number whose last digit is either 5 or 0, which must be labeled as alpha.
_________________
¿Tienes que presentar el GMAT y no sabes por dónde empezar? ¡Visita GMAT para Principiantes y recibe el curso completo gratis!




Re: Starting with 0, a mathematician labels every nonnegative integer as &nbs
[#permalink]
03 Sep 2018, 07:04






