What is the fewest number of marbles that Jeff could have?

Question

What is the fewest number of marbles that Jeff could have?
Here are the clues:
When he puts them in groups of 3, there are no marbles remaining.
When he puts them in groups of 2 or 5, there is one marble left.

Answer 1

Let x= fewest number of marbles that Jeff may have

Translate:

  "When he puts them in groups of 3, there are no marbles remaining."    means

         [x is number of marbles (dividend); remainder is 0; divisor "groups" is 3]  

           (x - 0)  =  (quotient1)*3

  "When he puts them into groups of 2 ... there is one marble left"     means

           (x - 1)  =  (quotient2)*2

  "When he puts them in groups of ... 5, there is one marble left."    means

           (x - 1)  =  (quotient3)*5

 

The math:

 

  Multiples of 3 (remainder 0 when divided by 3):      [note x must be one of these; pick smallest]

      3    6   9   12   15   18   21   24   27   30   33   36   39   42   45   48   51   54   57

  Remainder when divided by 3:

      0    0   0    0     0     0     0     0     0     0      0     0     0    0     0      0     0     0     0

  Remainder when divided by 2:

      1    0   1    0     1     0     1     0     1     0      1     0     1    0     1      0     1     0     1

  Remainder when divided by 5:

      3    1   4     2     0    3     1     4     2     0      3      1    4    2     0      3     1     4     2

 

The lowest value with a remainder of 0 when dividing by 3 and a remainder of 1 when dividing either by 2 or by 5 is 21.

 

Check: 

      21/3 = 7  rem 0

      21/2 = 10  rem 1

      21/5 = 4  rem 1

Answer 2

The least number of marbles is the least common multiple (LCM) of 2, 3, and 5.

 

2 · 3 · 5 = 30

 

The least number of marbles that Jeff can have is thirty.