dddanny2006 wrote:Is mn divisible by 40?
1) m is divisible by 10
2) n is divisible by 4
SOME BACKGROUND: A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N
Examples:
24 is divisible by
3 <--> 24 = (2)(2)(2)
(3)
70 is divisible by
5 <--> 70 = (2)
(5)(7)
330 is divisible by
6 <--> 330 =
(2)(3)(5)(11)
56 is divisible by
8 <--> 56 =
(2)(2)(2)(7)
Okay, now onto the question at hand . . .
Target question: Is mn divisible by 40?
Since 40 = (2)(2)(2)(5), we can rephrase the target question as . . .
REPHRASED target question: Are there three 2's and one 5 hiding in the prime factorization of mn?
Statement 1: m is divisible by 10
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 8, in which case
mn is divisible by 40
Case b: m = 10 and n = 3, in which case
mn is not divisible by 40
Since we cannot answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n is divisible by 4
There are several values of m and n that satisfy this condition. Here are two possible cases:
Case a: m = 10 and n = 4, in which case
mn is divisible by 40
Case b: m = 3 and n = 4, in which case
mn is not divisible by 40
Since we cannot answer the
target question with certainty, statement 2 is NOT SUFFICIENT
Statements 1 and 2 combined
Statement 1: Since 10 = (2)(5), statement 1 tells us that there is
one 2 and one 5 hiding in the prime factorization of m.
Statement 2: Since 4 = (2)(2), statement 2 tells us that there are
two 2's hiding in the prime factorization of n.
So, there must be
three 2's and one 5 hiding in the prime factorization of mn
Since we can answer the
target question with certainty, the combined statements are SUFFICIENT
Answer =
C
Cheers,
Brent
