Bunuel wrote:
If k is an integer and M and D are the least common multiple and the greatest common factor of 9k + 8 and 6k + 5, respectively, is M = 54k^2 + 93k + 40?
(1) D = 1
(2) D is a factor of 3k + 3
\(M = 54k^2 + 93k + 40=54k^2+48k+45k+40=(9k+8)(6K+5)\)..so the question asks : Is the LCM of these two numbers EQUAL to its product..
Now LCM*HCF= product of the two numbers, so HCF=D=1..
so we have to answer if D is 1?lets see the statements:-
(1) D = 1straightway gives us the answer..
suff
(2) D is a factor of 3k + 3.now is the slight tricky part
aggvipula) \((3k+3)*3 = 9k+9\) ....so 9k+9 and 9k+8 will have no factors in common except 1 as both are CONSECUTIVE integers
b) \((3k+3)*2 = 6k+6\) ....so 6k+6 and 6k+5 will have no factors in common except 1 as both are CONSECUTIVE integers
so the product (9k+8)(6K+5) wil laso have ONLY 1 as common factor with 3k+3..
But D is a common factor, so ONLY possible value is 1
suff
D
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