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.

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.

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

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.

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.