Let

*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

$\dfrac{9}{3} = B + \dfrac{2}{3}$
$B = \dfrac{7}{3}$ → (not a whole number - not okay)

Try *x* = 19

$\dfrac{19}{3} = B + \dfrac{2}{3}$
$B = \dfrac{17}{3}$ → (not a whole number - not okay)

Try *x* = 29

$\dfrac{29}{3} = B + \dfrac{2}{3}$
$B = 9$ → (whole number - okay)

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

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

Try *x* = 39

$\dfrac{39}{3} = B + \dfrac{2}{3}$
$B = \dfrac{37}{3}$ → (not a whole number - not okay)

Try *x* = 49

$\dfrac{49}{3} = B + \dfrac{2}{3}$
$B = \dfrac{47}{3}$ → (not a whole number - not okay)

Try *x* = 59

$\dfrac{59}{3} = B + \dfrac{2}{3}$
$B = 19$ → (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*