Determine the total number of concatenated 3-digit positive integers of the form ABC, such that each of their nonzero digits A, B, and C satisfies this relationship:
. . .
___ 0.A + 0.B + 0.C
0.ABC = ------------------
3Notes:
. ---
0.ABC = 0.ABCABCABC.......
.
0.A = 0.AAAA.......
Computer program/excel solver solutions are welcome- but a semi-analytic (p&p with a hand-calculator) methodology is preferred.
I'm assuming each of a,b,c is a single digit from 1 through 9.
(100a + 10b + c)/999 = (a+b+c)/27
(100a + 10b + c) = 37*(a+b+c)
63a = 27b + 36c
7a = 3b + 4c
The equation will always be true if a=b=c
If b is not equal to c, then b mod 7 = c mod 7.
So the only options for (b,c): {(1,8),(8,1),(2,9),(9,2)}
Each of these four options matches exactly one value for a.
Solutions:
(a,b,c) = (n,n,n) where n is in {1,2,3,4,5,6,7,8,9}
(a,b,c) = {(4,8,1), (5,1,8), (5,9,2), (6,2,9)}
Confirmed by a small computer program which outputs:
a b c
1 1 1
2 2 2
3 3 3
4 4 4
4 8 1
5 1 8
5 5 5
5 9 2
6 2 9
6 6 6
7 7 7
8 8 8
9 9 9
-----------------
for a in range(1,10):
for b in range(1,10):
for c in range(1,10):
if 7*a == 3*b + 4*c:
print(a,b,c)
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Posted by Larry
on 2023-08-01 06:36:26 |
Semi-analytic, confirmed by a program
