# Smallest number for given remainders

**Problem**

Find the smallest number which when divided by 2 the remainder is 1, when divided by 3 the remainder is 2, when divided by 4 the remainder is 3, when divided by 5 the remainder is 4, and when divided by 6 the remainder is 5.

**Solution**

*x*= the number

When divided by 2 the remainder is 1

$\dfrac{x}{2} = A + \dfrac{1}{2}$ → equation (1)

When divided by 3 the remainder is 2

$\dfrac{x}{3} = B + \dfrac{2}{3}$ → equation (2)

When divided by 4 the remainder is 3

$\dfrac{x}{4} = C + \dfrac{3}{4}$ → equation (3)

When divided by 5 the remainder is 4

$\dfrac{x}{5} = D + \dfrac{4}{5}$ → equation (4)

When divided by 6 the remainder is 5

$\dfrac{x}{6} = E + \dfrac{5}{6}$ → equation (5)

- From the above equations,
*x*,*A*,*B*,*C*,*D*, and*E*must be whole numbers. - From equation (1),
*x*must be odd. - From equation (4),
*x*must be divisible by 5 + the remainder 4.- If it ends with 0: 0 + 4 = 4 (even).
- If it ends with 5: 5 + 4 = 9 (odd)

Thus, *x* must end with 9.

Try *x* = 9

$B = \dfrac{7}{3}$ → (not a whole number - not okay)

Try *x* = 19

$B = \dfrac{17}{3}$ → (not a whole number - not okay)

Try *x* = 29

$B = 9$ → (whole number - okay)

$\dfrac{29}{4} = C + \dfrac{3}{4}$

$C = \dfrac{13}{2}$ → (not a whole number - not okay)

Try *x* = 39

$B = \dfrac{37}{3}$ → (not a whole number - not okay)

Try *x* = 49

$B = \dfrac{47}{3}$ → (not a whole number - not okay)

Try *x* = 59

$B = 19$ → (whole number - okay)

$\dfrac{59}{4} = C + \dfrac{3}{4}$

$C = 14$ → (whole number - okay)

$\dfrac{59}{6} = E + \dfrac{5}{6}$

$C = 9$ → (whole number - okay)

Thus, *x* = 59 *answer*

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