D={7, 14, 21,28,35,42,49} write in rule method
Answers
They're all multiples of 7, which means that they can each be represented as a number times 7. Or, because multiplication is really just a shortened form of addition, they can each be represented by a bunch of 7s being added together:
7 = 7 x 1 = 7
14 = 7 x 2 = 7 + 7
21 = 7 x 3 = 7 + 7 + 7
28 = 7 x 4 = 7 + 7 + 7 + 7
35 = 7 x 5 = 7 + 7 + 7 + 7 + 7
42 = 7 x 6 = 7 + 7 + 7 + 7 + 7 + 7
Now notice what happens when you try adding these numbers together. Let's say you add 21 and 28:
21 + 28 = (7 x 3) + (7 x 4) or (7 + 7 + 7) + (7 + 7 + 7 + 7)
The associative property of addition states that the grouping of elements doesn't make a difference; you can simply remove the parentheses when only addition is involved, which gives you this:
21 + 28 = 7 + 7 + 7 + 7 + 7 + 7 + 7 or 7 x 7
Since all multiples of 7 can be written as the sum of a certain number of 7s, whenever you add multiples of 7, the sum itself can also be written as the sum of a certain number of 7s, which is to say that if you add two or more multiples of 7, the sum is also a multiple of 7. This is true for all numbers; for example, if you add two or more multiples of 19, the sum is also a multiple of 19.
Looking back at the original problem, it is now clear that it's a trick question. Since you begin with all multiples of 7, there cannot be a subset of those numbers that sums to 100 because 100 is not a multiple of 7. The closest you can get is either 98 (42 + 35 + 21) or 105 (42 + 35 + 28).