What is the number?

Question

When split into two there is a remainder of 1. When split into 3 there is a remainder of 2. When split into 4 there is a remainder of 3. Thank you.

Answer 1

Divisor |  Remainder

------------------------

  2       |      1

  3       |      2

  4       |      3

 

Let the number we are looking for be X

Notice that each remainder is one less than the corresponding divisor.  

This means that X+1 will have a remainder of zero.

(X+1) must be divisible by 2, 3 & 4.

The simplest thing to do is multiply the three numbers:  2*3*4 =  24.

So, we have X+1 = 24.

X =23

 

Double-check the answer:   23/2 = 11 with a remainder of 1.   23/3 = 7 with a remainder of 2.   23/4 = 5 with a remainder of 3.

Answer 2

Multiples of 2:     2   4   6   8   10   12   14   16   18   20   22   24   26   28  30 32 34 ...

  +1                   3   5   7   9   11   13   15   17   19   21   23   25   27  29  31 33  35...

 

Multiples of 3:     3   6   9   12   15   18   21   24   27   30  33 ...

  +2                   5   8  11  14   17   20   23   26   28   32  35...

 

Multiples of 4:     4   8   12   16   20   24  28  32   ...

  +3                   7  11  15   19   23   27  31  35  ...

 

The number could be 11, or 23, or 35, or (11+12n , where n=0, 1, 2, 3, ...)

 

 

Note that the LCM of 2, 3, and 4 is 12.  That's where we get the 12; it is how often this happens.  We found 11 from lists, but it could have been calculated (Can you find out how?).