Check if you can solve the following:
The objective is to find a number which gets doubled when the last digit of the number is moved to the front of the number.
For an example:
Let abcd is the number, it should get doubled when d which is the last digit, is moved to the front, forming dabc
2 x abcd = dabc
can you find such a number or few numbers?
First I started building the following equation:
Let x is positive integer and y is a single digit.
x > 0 and 10 > y > 0
therefore, the number can be written as;
2 . (10 . x + y ) = (y . 10^p) + x
x = [ (10^p) – 2 ] * [y / 19]
y = [19 . x ] / [(10^p) – 2 ]
Then after, it is possible to solve this using trail and error method. It is cumbersome, yet possible. I was tying to figure out a heuristic method to get the answer, but I could not. Therefore I though of using a python script to get the answer.
<pre>#!/usr/bin/python def eqx(n, y): return int(((10**n)-(2))*(y/19)) def eqy(n, x): return int((19*x)/((10**n)-2)) def num_conv(num1, num2, p): d2 = num1%10 d1 = int(num1/10) if(num2==((d2*10**p)+d1) and (10**p<num1)): return True return False for i in range(0,300): for j in range(0,10): x = eqx(i,j) y = eqy(i,x) num1 = int((10*x)+y) num2 = int((y*(10**i))+x) LHS = 2*num1 RHS = num2 num_shuffled = num_conv(num1,num2,i) if ((LHS==RHS) and (num_shuffled) and (x!=0)): print('x,y,p',x,y,i) print (format(num1, ',d'),' shifting last digit ',format(num2, ',d')) print ('LHS',format(LHS, ',d'), ' RHS',format(RHS, ',d')) print('')</pre>
x,y,p 10526315789473684 2 17
105,263,157,894,736,842 shifting last digit 210,526,315,789,473,684
LHS 210,526,315,789,473,684 RHS 210,526,315,789,473,684
x,y,p 15789473684210526 3 17
157,894,736,842,105,263 shifting last digit 315,789,473,684,210,526
LHS 315,789,473,684,210,526 RHS 315,789,473,684,210,526
x,y,p 21052631578947368 4 17
210,526,315,789,473,684 shifting last digit 421,052,631,578,947,368
LHS 421,052,631,578,947,368 RHS 421,052,631,578,947,368
x,y,p 31578947368421052 6 17
315,789,473,684,210,526 shifting last digit 631,578,947,368,421,052
LHS 631,578,947,368,421,052 RHS 631,578,947,368,421,052
x,y,p 42105263157894736 8 17
421,052,631,578,947,368 shifting last digit 842,105,263,157,894,736
LHS 842,105,263,157,894,736 RHS 842,105,263,157,894,736
Later I found out these kind of numbers are called Parasitic numbers, OEIS sequence of A146088. If you are interested, refer to the following Wikipedia article as the starting point.