Henry Ernest Dudeney/Modern Puzzles/65 - Dividing by 37/Solution

by : $65$

 * Dividing by $37$

Solution

 * $49,129,308,213 = 37 \times 1,327,819,140 + 33$

Hence the remainder is $33$.

Proof
We proceed by Divisibility by 37.

In each group of $3$ digits of $49,129,308,213$, counting from the :
 * $\color {blue} {49}, \color {red} 1 \color {blue} {29}, \color {red} 3 \color {blue} {08}, \color {red} 2 \color {blue} {13}$

Then we add up the $\color {blue} {\text {blue} }$ numbers:

49 29 08 +13 --- 99

and subtract $11$ times the sum of the $\color {red} {\text {red} }$ numbers:

11 x (1 + 3 + 2) = 66

99 - 66 = 33

Hence the remainder is $33$.

To check, we perform the calculation using an online calculator or otherwise:
 * $49,129,308,213 = 37 \times 1,327,819,140 + 33$