Carcass wrote:
What is the lowest positive integer that is divisible by both 5,000,000 and 256?
A. 5,000,000
B. 10,000,000
C. 20,000,000
D. 50,000,000
E. 100,000,000
-----ASIDE---------------------
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 NConsider these examples:
24 is divisible by
3 because 24 = (2)(2)(2)
(3)Likewise, 70 is divisible by
5 because 70 = (2)
(5)(7)
And 112 is divisible by
8 because 112 = (2)
(2)(2)(2)(7)
And 630 is divisible by
15 because 630 = (2)(3)
(3)(5)(7)
-----ONTO THE QUESTION!---------------------
We can already see that all 5 answer choices are multiples of 5,000,000.
So it really comes down to which number is the smallest number that is divisible by 256.
256 = (2)(2)(2)(2)(2)(2)(2)(2)
So, the correct answer will be the smallest number that has eight 2's in its prime factorization
A) 5,000,000 = (5^7)(2^6), which means 5,000,000 has six 2's in its prime factorization
B) 10,000,000 = (5^7)(2^7), which means 5,000,000 has seven 2's in its prime factorization
C) 20,000,000 = (5^7)(2^8), which means 5,000,000 has EIGHT 2's in its prime factorization
Answer: C