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z is a positive integer and multiple of 2; p = 4z, what is the remaind
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25 Jun 2015, 04:42
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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25 Jun 2015, 05:48
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 NumberQuestion : 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 IntegerThe 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 Answer: Option
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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25 Jun 2015, 06:52
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 Answer B



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z is a positive integer and multiple of 2; p = 4z, what is the remaind
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25 Jun 2015, 07:22
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 Answer B.
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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25 Jun 2015, 09:02
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
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26 Jun 2015, 00:51
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
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26 Jun 2015, 07:33
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
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26 Jun 2015, 08:10
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
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26 Jun 2015, 09:50
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
Hence answer is B Thanks,



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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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27 Jun 2015, 03:02
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



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z is a positive integer and multiple of 2; p = 4z, what is the remaind
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29 Jun 2015, 05:21
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 Answer: B.
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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08 Aug 2016, 02:43
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.



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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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08 Aug 2016, 03:21



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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind
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08 Aug 2016, 03:31
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
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29 Jul 2018, 01:18
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Re: z is a positive integer and multiple of 2; p = 4z, what is the remaind &nbs
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