The smallest number which when diminished by 14, is exactly divisible 24, 32, 36, 42 and 56 is:
(A)2015
(B) 2030
(C) 2016
(D)2420
(A)2015
(B) 2030
(C) 2016
(D)2420
ASIDE: A lot of integer property questions can be solved using prime factorization.sgr21 wrote:The smallest number which when diminished by 14, is exactly divisible 24, 32, 36, 42 and 56 is:
(A)2015
(B) 2030
(C) 2016
(D)2420
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 to the question.
24 = (2)(2)(2)(3)
32 = (2)(2)(2)(2)(2)
36 = (2)(2)(3)(3)
42 = (2)(3)(7)
56 = (2)(2)(2)(7)
We need a number that it is divisible by 24, 32, 36, 42 and 56
That number will = (2)(2)(2)(2)(2)(3)(3)(7) = 2016 [notice that 24, 32, 36 etc are all "hiding in this prodcut]
So, 2016 is the smallest integer that's divisible by 24, 32, 36, 42 and 56
Of course, we're subtracting 14 from some number. So, the correct answer is 2030, since 2030 - 14 = 2016
Answer: B
Cheers,
Brent
