ABBCCA is a perfect square without a leading zero, and C=2B.
Find the letter to digit assignment.
511225 is the only square that fits the pattern.
I initially did a computer program, but Jer's method inspired me.
Let N be the square we are looking for.
N = 100001x + 11220y
x is in {1,4,5,6,9} since N is a square and x cannot be a leading zero.
y < 5 since C = 2B; y is in {0,1,2,3,4}
100001 = 11 * 9091
11220 = 2 * 2 * 3 * 5 * 11 * 17
x and y are both divisible by 11
N = 11*(9091x + 1020y)
Since N is a square, (9091x + 1020y) must have 11 as a factor.
9091 is 5 mod 11
1020 is 8 mod 11
5x + 8y must be 0 mod 11
(x,y) could be (0,0),(1,9),(2,7),(3,5),(4,3),(5,1),(6,10),(7,8),(8,6),(9,4)
But with restrictions on x and y: only (4,3),(5,1),(9,4)
All three yield N values that are divisible by 11^2,
but only (5,1) produces a perfect square.
511225
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Posted by Larry
on 2024-08-28 10:51:18 |
Longer analytic method
