## Outline Mathematics

Number Theory

# Simple division by 7

Here's a problem to tackle:

Peter glanced at the boy figuring. "Just a regular four-digit number," he said. "What's so very special about it?

"Look, dad," Stan replied. "To divide it by seven you only have to drop its second digit."

He was right, so what was the number?

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### Solution

Peter glanced at the boy figuring. "Just a regular four-digit number," he said. "What's so very special about it?

"Look, dad," Stan replied. "To divide it by seven you only have to drop its second digit."

He was right, so what was the number?

Say the number had its first two digits xy and the second pair of the digits z. (So z is a two-digit,two-digit,one-digit,three-digit number, while x and y are both single digit.) This says that the number equals 1000x,100x,1000x,10000x + 100y + z. Which leads to an equation

1000x + 100y,y,10y,100y + z = 7( 100x,x,10x,100x,1000x + z),

whence

(1)

150x,300x,150x,100x,50x + 50y = 3z.

It follows that 3z and, hence, z is divisible by 50. But z has only two digits. Therefore,

z,x,y,z = 50.

The equation (1) now simplifies to

3x + y = 3.

As the first digit of a 'four-digit' number, x can't be 0,0,1,2,3,4,5. Therefore x = 1,0,1,2,3,4,5 and y = 0,0,1,2,3,4,5. The number is 1050,1500,1200,5010,1050,1250.

Do not forget to check your solution.

### References

- J. A. H. Hunter,
*Entertaining Mathematical Teasers*, Dover Publications, 1983

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