If p.q and r are positive integers greater than 1, and p and q are factors of r, which of the following must be the factor of r^pq?
I. p+q
II. q^p
III. p^2 * q^2
A. I only
B. II only
C. III only
D. I and II
E. II and III
If pq and r are positive integers greater
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ANS:C… simply assume p=2 and q=3… would get answer…. but difficulty level is very less here……..
E
Since p & q are factors of r
hence p*Q*X=r ( let x be any number which is a factor of r )
so R^pq = (pqx)^pq= p^pq* q^pq * x^pq
1) cant be proven
2) Since R^pq = p^pq* q^pq * x^pq hence q^pq is a factor of r^pq
3) true
Answer is E
Let p=2,q=3 and r=6
If we go thru option wise only ii & iii satisfies the condition hence ‘E’
B
if we take
p = 2 q = 4 and r = 20
then iii does not satisfies
its only ii that satisfies the condition. So my answer is B
Ans is E
By question, one does comes to know that no matter what R is either greater thn or equal to p and q
@ Vikas Gupta, pls check your calculation once again. Cos even if we take r = 20, p = 2 and q = 4
Even then it satifies………
As ‘p’ and ‘q’ are factors of ‘r’,
r=pqx,where x is any other factor
r^pq=(pqx)^pq=p^pq×q^pq×x^pq
Option I does not hold true.
For option II, mid term in above expansion is q^pq=q^p q^q. Hence, it holds true.
For option III, since ‘p’ and ‘q’ are positive integers above 1, the value should be at least 2. Even then option III holds true.
Since option II and III holds true, the answer is E.
E
E
E
E
E
E
Coz the minimum vales that p and q can take are 2 and 2…
So, p^2 and q^2 will alwaz be present. So iii is true…
Also, q^p is true….so iii is also true
So E
E
E
THE ANSWER IS E
E
take an example:p=2,q=3,r=6
now p+q=5 is not factor of mentioned condition.
But, option II and III will be factor of mentioned condition.
E