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Question
1769 bottles of paint is arranged equally into 3, 4, 5, and 6 boxes. To leave the least number of bottles outside the boxes, the paint bottles must be divided equally into ____
1769 bottles of paint is arranged equally into 3, 4, 5, and 6 boxes. To leave the least number of bottles outside the boxes, the paint bottles must be divided equally into ____
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Solution
The correct option is B 4 boxes
Total number of paint bottles =1769
Now, 1769 paint bottles can be divided eqaully into any one of the following:
3 boxes, 4 boxes, 5 boxes or 6 boxes
The number of paint bottles left outside in each case is the remainder obtained by calculating 1769÷Number of boxes
Look for the least remainder.
So, if the bottles are packed equally into 3 boxes, calculate the remainder for 1769÷3
Calculating 1769÷3 by standard algorithm,
So, the remainder of 1769÷3 is 2
∴ If 1769 paint bottles are equally packed into 3 boxes, 2 bottles will remain outside the boxes.
Next, calculate the case in which the paint bottles are packed equally into 4 boxes.
Calculating 1769÷4 by standard algorithm,
So, the remainder of 1769÷4 is 1.
∴ If 1769 paint bottles are equally packed into 4 boxes, 1 bottle will remain outside the boxes.
Next, calculate the case where the paint bottles are equally packed into 5 boxes.
Calculating 1769÷5 by standard algorithm,
So, the remainder of 1769÷5 is 4.
∴ If 1769 paint bottles are equally packed into 5 boxes, 4 bottles will remain outside the boxes.
Next, calculate the case where the paint bottles are equally packed into 6 boxes.
Calculating 1769÷6,
So, the remainder of 1769÷6 is 4.
∴ If 1769 paint bottles are equally packed into 6 boxes, 5 bottle would remain outside the boxes.
Now, comparing the number of bottles remaining outside the boxes in each case,
Number of boxes the paint bottles are equally packed into
Number of bottles left outside
3 boxes
2 bottles
4 boxes
1 bottles
5 boxes
4 bottles
6 boxes
5 bottles
Comparing the number of bottles left outside, find that 1<2<4<5
∴ To leave the least number of paint bottles outside, the bottles must be arranged equally into 4 boxes.
Total number of paint bottles =1769
Now, 1769 paint bottles can be divided eqaully into any one of the following:
3 boxes, 4 boxes, 5 boxes or 6 boxes
The number of paint bottles left outside in each case is the remainder obtained by calculating 1769÷Number of boxes
Look for the least remainder.
So, if the bottles are packed equally into 3 boxes, calculate the remainder for 1769÷3
Calculating 1769÷3 by standard algorithm,
So, the remainder of 1769÷3 is 2
∴ If 1769 paint bottles are equally packed into 3 boxes, 2 bottles will remain outside the boxes.
Next, calculate the case in which the paint bottles are packed equally into 4 boxes.
Calculating 1769÷4 by standard algorithm,
So, the remainder of 1769÷4 is 1.
∴ If 1769 paint bottles are equally packed into 4 boxes, 1 bottle will remain outside the boxes.
Next, calculate the case where the paint bottles are equally packed into 5 boxes.
Calculating 1769÷5 by standard algorithm,
So, the remainder of 1769÷5 is 4.
∴ If 1769 paint bottles are equally packed into 5 boxes, 4 bottles will remain outside the boxes.
Next, calculate the case where the paint bottles are equally packed into 6 boxes.
Calculating 1769÷6,
So, the remainder of 1769÷6 is 4.
∴ If 1769 paint bottles are equally packed into 6 boxes, 5 bottle would remain outside the boxes.
Now, comparing the number of bottles remaining outside the boxes in each case,
| Number of boxes the paint bottles are equally packed into | Number of bottles left outside |
| 3 boxes | 2 bottles |
| 4 boxes | 1 bottles |
| 5 boxes | 4 bottles |
| 6 boxes | 5 bottles |
Comparing the number of bottles left outside, find that 1<2<4<5
∴ To leave the least number of paint bottles outside, the bottles must be arranged equally into 4 boxes.
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