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 ____

3 boxes

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4 boxes

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5 boxes

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6 boxes

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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:
$3 b o x e s$ , $4 b o x e s$ , $5 b o x e s$ or $6 b o x e s$

The number of paint bottles left outside in each case is the remainder obtained by calculating $1769 \div N u m b e r o f b o x e s$

Look for the least remainder.

So, if the bottles are packed equally into $3$ boxes, calculate the remainder for $1769 \div 3$

Calculating $1769 \div 3$ by standard algorithm,

So, the remainder of $1769 \div 3$ 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 $1769 \div 4$ by standard algorithm,

So, the remainder of $1769 \div 4$ 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 $1769 \div 5$ by standard algorithm,

So, the remainder of $1769 \div 5$ 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 $1769 \div 6$ ,

So, the remainder of $1769 \div 6$ 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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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