Question 96633: There are less than 100 balls in a box. If you remove the balls in threes, 1 ball will remain in the box. If you remove the balls in fours, 2 balls will remain. If you remove the balls in fives, 3 balls will remain in the box. If your remove the balls in sixs, four balls will remain.
How many balls are there in the box?
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! The answer is 58 balls are in the box.
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Probably the easiest way to do this problem is by using intuition to eliminate possible answers.
The problem tells you that there are 99 possible answers (1 through 99) and one way to find
the answer is to start with 99 and find out whether you can divide it by 6 and have a remainder
of 4. If that works, then divide 99 by 5 and see if the remainder is 3. If that works, then
divide 99 by 4 and see if the remainder is 3, and if that works, divide 99 by 3 and see if
the remainder is 1.
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That sure seems like a lot of work. So let's see how we might trim it down.
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Let's begin by working in reverse. Let's start with the number of 6. If we take multiples of 6
and add 4 to each of them, we get 15 possible answers:
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6 times 1 = 6 + 4 = 10
6 times 2 = 12 + 4 = 16
6 times 3 = 18 + 4 = 22
6 times 4 = 24 + 4 = 28
6 times 5 = 30 + 4 = 34
6 times 6 = 36 + 4 = 40
6 times 7 = 42 + 4 = 46
6 times 8 = 48 + 4 = 52
6 times 9 = 54 + 4 = 58
6 times 10 = 60 + 4 = 64
6 times 11 = 66 + 4 = 70
6 times 12 = 72 + 4 = 76
6 times 13 = 78 + 4 = 82
6 times 14 = 84 + 4 = 88
6 times 15 = 90 + 4 = 94
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The right side of this table shows those numbers that if you divide them by 6 the remainder
is 4. So there are only 15 numbers that can be answers to this problem.
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Next let's look at the possible answers insofar as 5 is involved. Multiples of 5 end in only
two numbers - zero and 5. When you add 3 (the remainder) to these possibilities, the answer
will either end in 3 or in 8. Now you look down the table for the sixes, and you find
that of the possible answers for the sixes none of them end in 3 and only three end in 8
... 28, 58, and 88. So only 28, 58, and 88 will satisfy the requirements that if you divide
them by 6 you will get a remainder of 4 AND if you divide them by 5 the remainder will be 3.
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Next let's check these three numbers by seeing if they satisfy the requirement that if they
are divided by 4 the remainder is 2. Divide 28 by 4 and the answer is 7 with no remainder.
Divide 58 by 4 and the answer is 14 with a remainder of 2. Finally, divide 88 by 4 and the
answer is 22 with no remainder. At this point we know that the only possible answer is 58.
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Check this answer by dividing 58 by 3 and seeing if the remainder is 1. If you divide 58 by 3
the answer is 19 with a remainder of 1. So this checks and confirms the answer we found.
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So we found the answer by finding a way to narrow the number of possibilities down to a manageable
amount and eliminate a lot of work.
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Hope this helps you to think of other things that might be done to solve this problem and
at least to understand that the above possibility exists.
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