Question 1208789: Given the set {1,2,3,6,12,23,44), which integers between 60 and 80 cannot be written as a number from the set on its own, or as the sum of less than five numbers from the given set?
Answer by greenestamps(13203)   (Show Source):
You can put this solution on YOUR website!
It seems likely to me that it will be easier to determine which numbers between 60 and 80 you CAN make as the sum of fewer than 5 of the numbers in the given set.
The sum of the four largest numbers in the set other than 44 is less than 60, so the number 44 must always be included.
Suppose we use both 44 and 23; that sum is 67. So to check to see if we can make sums of the numbers 67 to 80, we need to see if we can make sums of 0 to 13 using no more than two of the other numbers.
0: 44+23 = 67
1: 44+23+1 = 68
2: 44+23+2 = 69
3: 44+23+3 = 70
4 = 3+1: 44+23+3+1 = 71
5 = 3+2: 44+23+3+2 = 72
6: 44+23+6 = 73
7 = 6+1: 44+23+6+1 = 74
8 = 6+2: 44+23+6+2 = 75
9 = 6+3: 44+23+6+3 = 76
76 is the largest number we can make as the sum of fewer than 5 of the given numbers if we use 44 and 23 and don't use the number 12. So the next smallest number we can make as the sum of fewer than 5 of the given numbers is 44+23+12 = 79; and then we can also make 44+23+12+1 = 80.
So two of the numbers we can NOT make as the sum of fewer than 5 of the given numbers are 77 and 78.
We have determined all the numbers greater than 67 that we can and can't make as the sum of fewer than 5 of the given numbers. Now what about making numbers less than 67....
We have to use 44; and we can't use 23. So we look for combinations using 44 and 12 as the two largest numbers from the given set. 44+12 = 56; to make sums of 60 to 66, we are looking for ways to make the sums 4 to 10 using at most two of the remaining numbers.
4 = 3+1: 44+12+3+1 = 60
5 = 3+2: 44+12+3+2 = 61
6: 44+12+6 = 62
7 = 6+1: 44+12+6+1 = 63
8 = 6+2: 44+12+6+2 = 64
9 = 6+3: 44+12+6+3 = 65
65 is the largest sum we can make using 44 and 12 and at most two of the other numbers. So we can make sums of all the numbers from 60 to 65 except for 66.
So the one other number between 60 and 80 that we can't make as the sum of at most 4 of the given numbers is 66.
ANSWER: The numbers we can't make as the sum of no more than 4 of the given numbers are 66, 77, and 78.
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