M is the least common multiple of 90,196 and 300. Which of the following is the factor of M
A) 600
B) 700
C) 900
D) 2100
E) 4900
OAA
A) 600
B) 700
C) 900
D) 2100
E) 4900
OAA
To find the LCM of three numbers, use the prime factorization:
90 = 9*10 = 2*(3^2)*5
196 = 200 - 5 = 49*4 = (2^2)(7^2)
300 = 3*100 = (2^2)*3*(5^2)
You're supposed to use the highest powers of every prime factor, so LCM will be (2^2)*(3^2)*(5^2)*(7^2).
Now, given the OA, I'm tempted to say that the question is actually asking for the number that IS NOT a factor of the LCM of the 3 numbers. In this case, 600 would indeed be it, since 600 = 6*100 = 2*3*(2^2)*(5^2) = (2^3)*3*(5^2). As you can see, the fact that 2 is in the third power is actually the issue here, since we don't have a 2 to the third power in the LCM.
The prime factorization of the rest of your options proves that they are all factors if the LCM:
B. 700 = (2^2)*(5^2)*7
C. 900 = (2^2)*(3^2)*(5^2)
D. 2100 = (2^2)*3*(5^2)*7
E. 4900 = (2^2)*(5^2)*(7^2).
The trick is to notice that the powers of all the prime factors of say 700 are less than those of the LCM.
