**Concatenation** is the joining of two numbers by their numerals. That is, the concatenation of **444** and **777** is **444777**.

Concatenation of numbers **A** and **B** is denoted **A || B**.

**A** and **B** are three-digit integers. So **A || B** is a 6-digit number.

So, what is the smallest integer of the form (A || B) / B ?

Note the trivial solution is :

A = 001, B = 500, then A || B = 001500 and (A || B)/B = 001500/500 = 3

### Like this:

Like Loading...

*Related*

## About benvitalis

math grad - Interest: Number theory

#!/usr/bin/python

# David a/k/a/ @InfinitelyManic

# Concatenation puzzle: when (A||B)/B is an integer

# import Fractions from fractions module

from fractions import *

# declare list sol[utions]

sol = []

# we are going to put the data in a file

f = open(‘results.txt’,’w’)

# create two (2) for loops using range. Range will end at 999, respectively.

for A in range(100,1000):

for B in range(100,1000):

# concat A & B; turn into an int then do the math

frac = Fraction(int(str(A)+str(B)),int(B))

# if the denominator is 1 then write “A,B, fraction form, and numerator” to text file

if frac.denominator == 1:

sol = [A,B,str(A)+str(B),’/’,B,frac.numerator]

f.write(str(sol)+’\n’)

# close the file

f.close()

# #########Linux Bash ##############

$ sort -n -k6 -t, results.txt | less # using sort to sort numerator value within file to find target numerator value (i.e., smallest & largest)