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