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

Modern Puzzles by Henry Ernest Dudeney: $65$

Dividing by $37$

I want to know whether the number $49,129,308,213$ is exactly divisible by $37$,

or if not, what is the remainder when so divided.

How may I do this quite easily without any process of actual division whatever?

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 least significant digit:

$\color {blue} {49}, \color {red} 1 \color {blue} {29}, \color {red} 3 \color {blue} {08}, \color {red} 2 \color {blue} {13}$

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:

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

$\blacksquare$

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Solutions: $65$. -- Dividing by $37$
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Answers: Digital Puzzles: $116$. Dividing by $37$