bhandariavi wrote:
:oops:
If k is a positive integer, is K a prime number?
1) No integers between "2" and "square root of K" inclusive divides k evenly?
2) No integers between 2 and k/2 divides k evenly, and k is greater than 5.
I will post the official answer later on.
Thanks,
Avi.
Given:
k=integer>0. Question:
k=prime?
A prime number is a positive integer with exactly two distinct divisors: 1 and itself.
(1) No integer between 2 and
\sqrt{k} inclusive divides k evenly --> let's assume
k is not a prime, then there must be some integers
a and
b (
1<a<k and
1<a<k), a factors of
k, for which
ab=k. As given that
k has no factor between 2 and
\sqrt{k} inclusive, then both factors
a and
b must be more than
\sqrt{k}. But it's not possible, as the product of two positive integers more than
\sqrt{k} will yield an integer more than
k (
ab>k). Hence our assumption that
k is not a prime is not true -->
k is a prime. Sufficient.
(2) No integers between 2 and
\frac{k}{2} divides k evenly, and k is greater than 5 --> the same here : let's assume
k is not a prime, then there must be some integers
a and
b (
1<a<k and
1<a<k), a factors of
k, for which
ab=k -->
k=ab\geq{\frac{k^2}{4}} (as both
a and
b are more than or equal to
\frac{k}{2}, then their product
ab, which is
k, must be more than or equal to
\frac{k}{2}*\frac{k}{2}) -->
4k\geq{k^2} -->
k(4-k)\geq{0}. But this inequality cannot be true as
4-k will be negative (as given
k>5) and
k is positive so
k(4-k) must be negative not positive or zero. Hence our assumption that
k is not a prime is not true -->
k is a prime. Sufficient.
Answer: D.
P.S. The first statement is basically the way of checking whether some # is a prime:
Verifying the primality (checking whether the number is a prime) of a given number
n can be done by trial division, that is to say dividing
n by all integer numbers (primes) smaller than
\sqrt{n}, thereby checking whether
n is a multiple of
m<\sqrt{n}.
Example: Verifying the primality of
161:
\sqrt{161} is little less than
13, from integers from
2 to
13,
161 is divisible by
7, hence
161 is not prime.
Hope it helps.
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