What 3-dgit number abc
squared divides abcabc
NB: italic font denotes a concatenation, as opposed to a product.
Note: abcabc=1001 x abc
The prime factors of 1001 are 7, 11, 13.
The factors of 1001 are: 7,11,13,91,143
Since 1001 = f1 x c1 where f1 is a factor and c1 is its co-factor,
f1 x 1001 / f1^2 = c1
Each factor yields a solution, but only c1=7, f1= 143 yields a>0
e.g.: 143143/143^2=7, while 091091/91^2 = 11
Note also that this doesn't work again soon. That is, amongst 101, 1001, 10001, 100001, etc., 1001 is unusual to have a factor, 143, with only one less digit. Many of the others are prime or composite of only two primes.
rabbit-3:~ lord$ sqq
143143 / 143^2 = 7.000
rabbit-3:~ lord$ more sqq.f
if(a.eq.b)go to 3
if(c.eq.a.or.c.eq.b)go to 2
1 format(i6,' / ',i3,'^2 =',f8.3)
(There are no additional solutions when a,b,c are allowed to be non-unique, so long as a>0)
Edited on September 22, 2018, 10:06 am