Discussion
Which one of the following could be the difference between two numbers both of which are divisible by 2, 3 and 4?
*This question is included in Nova Math - Problem Set F: Number Theory, question #24
| (A) | 71 |
| (B) | ... |
| (C) | ... |
| (D) | ... |
| (E) | ... |
| (F) | ... |
The solution is
Posted: 07/27/2013 08:42
Can you please explain what (difference between two numbers) means in here?
Posted: 07/27/2013 10:57
Sudad, let's say two numbers are a and b, and a > b. The difference between the two numbers is a - b.
Posted: 12/03/2015 19:57
In the list given, only 72 is divisible by all three numbers. What why does the 12 have to come into the equation? Shouldn't the problem be over once you discover only 72 is divisible by them?
Posted: 12/03/2015 20:42
Hi Joelene,
Although each of the two numbers is divisible by 2, 3, and 4, we did not know whether the difference of the two numbers was also divisible by 2, 3, and 4. We had to prove that, and did so by showing that their difference is divisible by 12, which is same as saying the difference is divisible by 2, 3, and 4 since the factors of 12 are 2, 3, and 4.
Note: Instead of using the 12, we can carry the factors 2, 3, and 4 throughout the proof (although this makes the proof messy, perhaps it's clearer):
Let the first number be represented as (2)(3)(4)a and the second number as (2)(3)(4)b. Assuming a > b, the difference between the two numbers is (2)(3)(4)a − (2)(3)(4)b = (2)(3)(4)(a − b). Observe that this number is also a multiple of (2)(3)(4). Hence, the answer must also be divisible by (2)(3)(4). Since 72 is the only answer-choice divisible by (2)(3)(4), the answer is (B).
Nova Press
Although each of the two numbers is divisible by 2, 3, and 4, we did not know whether the difference of the two numbers was also divisible by 2, 3, and 4. We had to prove that, and did so by showing that their difference is divisible by 12, which is same as saying the difference is divisible by 2, 3, and 4 since the factors of 12 are 2, 3, and 4.
Note: Instead of using the 12, we can carry the factors 2, 3, and 4 throughout the proof (although this makes the proof messy, perhaps it's clearer):
Let the first number be represented as (2)(3)(4)a and the second number as (2)(3)(4)b. Assuming a > b, the difference between the two numbers is (2)(3)(4)a − (2)(3)(4)b = (2)(3)(4)(a − b). Observe that this number is also a multiple of (2)(3)(4). Hence, the answer must also be divisible by (2)(3)(4). Since 72 is the only answer-choice divisible by (2)(3)(4), the answer is (B).
Nova Press