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# z is a positive integer and multiple of 2; p = 4z, what is the remaind

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Math Expert
Joined: 02 Sep 2009
Posts: 50613
z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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25 Jun 2015, 03:42
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Difficulty:

15% (low)

Question Stats:

76% (01:21) correct 24% (01:21) wrong based on 149 sessions

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z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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25 Jun 2015, 04:48
Bunuel wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

CONCEPT: Remainder when any Number is divided by 10 is same as the Unit digit of the Number

Question : What is the remainder when p = 4^z is divided by 10? is same as

Question : What is the Unit Digit of p = 4^z?

Given: z is an even Integer

The cyclicity of Unit Digit of 4 is 2 i.e. Unit digit of powers of 4 repeat after every two powers in the pattern {4, 6, 4, 6, 4, ....}
$$4^1 = 4$$
$$4^2 = 16$$
$$4^3 = 64$$
$$4^4 = 256$$

i.e. Every Even Power of 4 gives the Unit digit = 6 = Remainder

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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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25 Jun 2015, 05:52
1
Bunuel wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

Let z=2 or 4 or 6 etc
Then, p=4^2 or 4^4 or 4^6
Now, 4^2=16 and this divided by 10 gives a remainder of 6
4^4=256 and this divided by 10 also gives a remainder of 6
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z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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25 Jun 2015, 06:22
1
Bunuel wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

If p = 4^2, the remainder of 16/10 is 6
If p = 4^4, the remainder of 256/10 is 6

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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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25 Jun 2015, 08:02
1
Since every number elevated at the at a series of exponents shows a pattern, in this case we spot it from the first three number tested (numbers allowed by the constrains):

- 4^2=16
- 4^3=256
- 4^4=4096

The reminder is always 6.
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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25 Jun 2015, 23:51
1
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Solution -

z is a positive integer and multiple of 2 -> z=2n

Substitute z in the equation -> p=4^z = 4^2n = 16^n

For any any value of positive integer n, unit digit of p = 6 . Hence the remainder when divided by 10.

ANS. B

Thanks
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z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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26 Jun 2015, 06:33
1
Bunuel wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

Z is positive and multiple of 2
Assume Z = 2K

P = 4^Z = $$([m](2^2)$$) ^ 2K[/m] = $$2^(4K)$$

As per the cyclicity of 2
2^1 -> unit digit 2
2^2 -> unit digit 4
2^3 -> unit digit 8
2^4 -> unit digit 6

Hence P will have unit digit as 6, Hence when divided by 10 will leave remainder as 6

Option B
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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26 Jun 2015, 07:10
1
I will pick two numbers 2 and 4. When I do I get 6 . Answer B
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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26 Jun 2015, 08:50
Z =4,6,8,10……..
P= 4^z ;
4^1 =4
4^2=16
4^3 =64
4^4 = ..6 (unit digit will be 6)
Likewise the pattern shows that even power have unit digit as 6 and odd power has unit digit as 4.
Hence 4^(even power) = unit digit =6; and 6/10 =reminder =6

Thanks,
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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27 Jun 2015, 02:02
1
Bunuel wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

p = 2^2z .. cycle of 2: 2, 4, 8 , 6

2^2z will always have 6 as the last digit as it will always have some power of 2^2 in it.. Answer B
Math Expert
Joined: 02 Sep 2009
Posts: 50613
z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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29 Jun 2015, 04:21
Bunuel wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Source: Platinum GMAT
Kudos for a correct solution.

Platinum GMAT Official Solution:

It is essential to recognize that the remainder when an integer is divided by 10 is simply the units digit of that integer. To help see this, consider the following examples:
4/10 is 0 with a remainder of 4
14/10 is 1 with a remainder of 4
5/10 is 0 with a remainder of 5
105/10 is 10 with a remainder of 5

It is also essential to remember that the z is a positive integer and multiple of 2. Any integer that is a multiple of 2 is an even number. So, z must be a positive even integer.

With these two observations, the question can be simplified to: "what is the units digit of 4 raised to an even positive integer?"

The units digit of 4 raised to an integer follows a specific repeating pattern:
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256
4^(odd number) --> units digit of 4
4^(even number) --> units digit of 6

There is a clear pattern regarding the units digit. 4 raised to any odd integer has a units digit of 4 while 4 raised to any even integer has a units digit of 6.

Since z must be an even integer, the units digit of p=4^z will always be 6. Consequently, the remainder when p=4^z is divided by 10 will always be 6.

In case this is too theoretical, consider the following examples:
z=2 --> p=4^z=16 --> p/10 = 1 with a remainder of 6
z=4 --> p=4^z=256 --> p/10 = 25 with a remainder of 6
z=6 --> p=4^z=4096 --> p/10 = 409 with a remainder of 6
z=8 --> p=4^z=65536 --> p/10 = 6553 with a remainder of 6

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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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08 Aug 2016, 01:43
Can´t we simplify 4^z to 2^2z and then solve accordingly? I know that not simplifying is better but it just arised in my mind and solving accordingly i:e; by simplifying I did not get the answer.
Want explanation from experts like Bunuel.
Math Expert
Joined: 02 Sep 2009
Posts: 50613
Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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08 Aug 2016, 02:21
NaeemHasan wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Can´t we simplify 4^z to 2^2z and then solve accordingly? I know that not simplifying is better but it just arised in my mind and solving accordingly i:e; by simplifying I did not get the answer.
Want explanation from experts like Bunuel.

Yes, $$4^z=2^{2z}$$. Since z is even we can further write this as $$2^{2*2k}=2^{4k}$$.
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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08 Aug 2016, 02:31
Bunuel wrote:
NaeemHasan wrote:
z is a positive integer and multiple of 2; p = 4^z, what is the remainder when p is divided by 10?

A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined

Can´t we simplify 4^z to 2^2z and then solve accordingly? I know that not simplifying is better but it just arised in my mind and solving accordingly i:e; by simplifying I did not get the answer.
Want explanation from experts like Bunuel.

Yes, $$4^z=2^{2z}$$. Since z is even we can further write this as $$2^{2*2k}=2^{4k}$$.

Thank you for your awesome explanation.
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind  [#permalink]

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29 Jul 2018, 00:18
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind &nbs [#permalink] 29 Jul 2018, 00:18
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