Is x the square of an integer?
(1) x = 12k + 6, where k is a positive integer
(2) x = 3q + 9, where q is a positive integer
Questions asks whether x=I^2 where I is a Integer.
from St 1 we have that x= 12K+6 or x= 6(2K+1)
Now when can 6*(2k+1) will be square??
2k+1 has to equal to 6 or 24 (6*2*2) or 54 (6*3*3) or 96 (6*4*4) to make 6*(2k+1) as square of integer.
But if 6=2k+1 then k =5/2 which is not an integer and is the case for all the values as well.
Hence St 1 is sufficient and option B,C and E ruled out
St 2 we have x=3q+9 or x=3 (q+3) now Q is a positive integer and for values of q =9, x=36 ie. square of an integer and if q= 1,x=12 ie. not a square of an integer.
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