If both 112 and 33 are factors of the number a * 43 * 62 * 1311, then what is the smallest possible value of a?  Choices.
A) 121 B) 3267 C) 363 D) 33 E) none of the above

Rationale: explanatory answer step 1 : prime factorize the given expression a * 43 * 62 * 1311 can be expressed in terms of its prime factors as a * 28 * 32 * 1311 step 2 : find factors missing after excluding ' a ' to make the number divisible by both 112 and 33 112 is a factor of the given number . if we do not include ' a ' , 11 is not a prime factor of the given number . if 112 is a factor of the number , 112 should have been in ' a ' 33 is a factor of the given number . if we do not include ' a ' , the number has only 32 in it . therefore , if 33 has to be a factor of the given number ' a ' has to contain 31 in it . therefore , ' a ' should be at least 112 * 3 = 363 if the given number has 112 and 33 as its factors . the smallest value that ` ` a ' ' can take is 363 . choice c is the correct answer .
Answer: C