There are between 20 and 30 marbles in a jar. If they are counted out 4...

Question

There are between 20 and 30 marbles in a jar. If they are counted out 4 at a time, there are 3 left over, but if they are counted out 5 at a time, there are 2 left over. How many marbles are in the jar?

GMAT Panda · Accepted Answer

27 marbles We define the variables and translate the first remainder condition "If they are counted out 4 at a time, there are 3 left over". n = number of marbles n = 4k + 3 We translate the second remainder condition "if they are counted out 5 at a time, there are 2 left over". m = number of full groups of 5 n = 5m + 2 We list all values between 20 and 30 that satisfy the first condition n = 4k + 3. n = 4k + 3 n 23, 27 We test n = 23 in the second condition and rule it out because it leaves the wrong remainder. 23 = 5 4 + 3 23 We test n = 27 in the second condition and confirm it leaves the correct remainder of 2. 27 = 5 5 + 2

Explanation	Calculations	Help
$We define the variables and translate the first remainder condition "If they are counted out 4 at a time, there are 3 left over" .$	$n = number of marbles n = 4 k + 3$
$We translate the second remainder condition "if they are counted out 5 at a time, there are 2 left over" .$	$m = number of full groups of 5 n = 5 m + 2$
$We list all values between 20 and 30 that satisfy the first condition n = 4 k + 3 .$	$n = 4 k + 3 n \in {23, 27}$
$We test n = 23 in the second condition and rule it out because it leaves the wrong remainder.$	$23 = 5 \times 4 + 3 23$
$We test n = 27 in the second condition and confirm it leaves the correct remainder of 2.$	$27 = 5 \times 5 + 2$

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