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# Starting with 0, a mathematician labels every non-negative integer as

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Math Expert
Joined: 02 Sep 2009
Posts: 49271
Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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14 Sep 2015, 23:04
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65% (02:16) correct 35% (02:38) wrong based on 106 sessions

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Starting with 0, a mathematician labels every non-negative 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.

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Posts: 2314
Location: India
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Re: Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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14 Sep 2015, 23:52
Bunuel wrote:
Starting with 0, a mathematician labels every non-negative 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"

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Re: Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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15 Sep 2015, 02:26
3
Bunuel wrote:
Starting with 0, a mathematician labels every non-negative 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
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Posts: 171
Location: United States
GPA: 3.5
WE: Engineering (Telecommunications)
Re: Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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15 Sep 2015, 20:17
1
1
Starting with 0, a mathematician labels every non-negative 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
1--B beta
2--G gamma
3--D delta
4--E epsilon
5--A
6--B
7--G
8--D
9--E
10--A

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 .
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Posts: 159
Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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17 Sep 2015, 13:53
1
Bunuel wrote:
Starting with 0, a mathematician labels every non-negative 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------Beta------Gamma------Delta------Epsilon------

--------0-----------1------------2-------------3-----------4-------

--------5-----------6------------7-------------8-----------9-------

--------10---------11----------12------------13----------14-------

--------15---------16----------17------------18----------19-------

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.

Math Expert
Joined: 02 Sep 2009
Posts: 49271
Re: Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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20 Sep 2015, 21:09
Bunuel wrote:
Starting with 0, a mathematician labels every non-negative 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 non-multiples 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.

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Re: Starting with 0, a mathematician labels every non-negative integer as  [#permalink]

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