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
