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K^4 is divisible by 32. What is the remainder when K is

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K^4 is divisible by 32. What is the remainder when K is [#permalink] New post 12 May 2005, 07:48
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E

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K^4 is divisible by 32. What is the remainder when K is divided by 32?
1) The remainder is 3 when K is divided by 4
2) The remainder is 2 when K is divided by 3
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 [#permalink] New post 12 May 2005, 09:18
Nightmare.
I'd pick E and give up.
First one doesn't make sense at all.

K^4 is divisable by 32=2^5
(1) K = 4m + 3

K^4 =(4m + 3)^4 = (4m)^4 + 4 * (4m)^3 * 3 + 6 * (4m)^2 * 3^2 + 4 * (4m) * 3^3 + 3^4

First two are divisable by 32 => 4*(4m)*3^3 + 3^4 = 3^3 (4*(4m) + 3)) - divisable by 32 => 4*(4m) + 3 is divisable by 32 => 16m = 29 (mod 32)

You just can't find an integer n so that 16m = 32n + 29 => 16(m-2n) = 29, 16 and 29 have no common divisors at all.

this does not hold. You can't find a number so that k^4 = 0 (mod 2^5) and k = 3 (mod 2^2)

Out.
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 [#permalink] New post 12 May 2005, 23:51
Tyr wrote:
Nightmare


I definitely agree...it's not the first time that I see this exact problem but I still can not solve it ...shame on me :oops:
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 [#permalink] New post 13 May 2005, 14:30
is this really going to appear in the test? my friend wrote a very complicated solution which for sure cannot take 2-3 minutes. his answer was also E. if you are interested i can post it.
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 [#permalink] New post 17 May 2005, 19:35
E
Note:- The first one does not make sense as there is no value of K that can fulfill the given condition as well as option 1 condition.
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 [#permalink] New post 17 May 2005, 22:33
1st statement indicates that K is odd but our problem states that
((k/2)^4)*(1/2). That is, K must be even to have this statement. So, statement contradicts with the problem. I guess it can't be sufficient. It's something different :shock:

2nd statement is insufficient. When k=8 (it also satisfies original statement) the remainder is 1. When k=20, the remainder is 5.

Combining statements won't suffice either. 1st one is weird :?:

Answer is E. july05, please post the explanation of your friend. Thanks.
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answer is B :D <-- this is smilly not d :d [#permalink] New post 17 May 2005, 23:35
this question is bit diffy not one of the most diffy questions those who r targetting 700+ can xpect even harder questions in last five Qs :)
anyways coming back to the solution from statment B K can b 5,8 and so on as 5 do not fit in orginal statment so it is 8 then 8*8*8*8 leaves remainder as 0, then it cant b 11, 14.... (x*3+2 where x can b from 1...32)
32 may work but i havent checked till tht value.
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i believe that all the problem solving calculations in Quantitative section have simple calculations. No Long Mahcine requiring calculations r required :D

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 [#permalink] New post 18 May 2005, 00:20
here it is. i pasted this from an email, so the power sign "^" was gone. i tried to fix it, but if you find any more inconsistences like that, please let me know.

---------------

If you are still interested, you can refer to K. Rosen's "Discrete Mathematics," the part dealing with modular arithmetic.

The original statement
K^4 = 0 mod 32 (K^4 is divisible by 32)
can be reduced to
K = 0 mod 4 (K is divisible by 4)

Let's look at this closely.
K^4 = n * 32
K^4 = n * 2^5
K^4 = 2n * 2^4
K = rt4(2n) * 2

Since K is an integer, rt4(2n) must also be an integer. This is the case
only when n is divisible by 8, let n = 8m.
K = rt4(2*8m) * 2 = rt4(16m) * 2 = 2 * rt4(m) * 2 = 4 * rt4(m)

In this expression rt4(m) may be any integer. In other words, K = 4x. This is sufficient and necessary condition on K. The statements K=4x and K^4 is divisible by 32 are equivalent.

Let us proceed. Original statement in its new form K=4x contradicts (1).
Therefore we should not consider (1). Let us look into (2). Taken alone, (2) suggests the number set K = 2+3y: 2, 5, 8, 11, 14, etc.

Let us combine (2) with our original condition of K=4x:
K = 4x
K = 2+3y

Since 3 and 4 are mutually prime, by some theorem (maybe it's Chinese
Remainder Theorem, i'm not sure), the solution to this equation set is a
number range with a period of 3*4 = 12. The first number satisfying both
conditions is 8. All the numbers can be represented as K = 8 + 12z, where z
is some integer.

They are 8, 20, 32, 44, etc.

These numbers have different remainders by 32, for example 8 has 8, 20 has
20, and 32 has 0. It means that (2) is insufficient.

(1) and (2) taken together contradict the (0) condition because (1)
contradicts (0).

The answer is (E). Neither (1), nor (2) is sufficient.
  [#permalink] 18 May 2005, 00:20
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