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 Six Seven Eight (Posted on 2005-10-05)
The letters E, G, H, I, N, S, T, V, and X represent different digits 0-9 (Neither S nor E is 0). SIX is a three digit number equal to the product of two consecutive integers, SEVEN is a five digit prime number, and EIGHT is a five digit perfect cube. What numbers are SIX, SEVEN and EIGHT?

 See The Solution Submitted by Brian Smith No Rating

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 Computer solution -- spoiler | Comment 2 of 5 |

The following program only checks SEVEN for divisibility by primes up to 19, but that narrows it down sufficiently to test the primality of SEVEN by other means:

DIM used(9)

FOR s = 1 TO 9
used(s) = 1
FOR i = 0 TO 9
IF used(i) = 0 THEN
used(i) = 1
FOR x = 0 TO 9
IF used(x) = 0 THEN
used(x) = 1
six = 100 * s + 10 * i + x
sr1 = INT(SQR(six))
IF sr1 * (sr1 + 1) = six THEN
FOR e = 1 TO 9
IF used(e) = 0 THEN
used(e) = 1
FOR g = 0 TO 9
IF used(g) = 0 THEN
used(g) = 1
FOR h = 0 TO 9
IF used(h) = 0 THEN
used(h) = 1
FOR t = 0 TO 9
IF used(t) = 0 THEN
used(t) = 1
eight = (((e * 10 + i) * 10 + g) * 10 + h) * 10 + t
cr = INT(eight ^ (1 / 3) + .5)
IF cr * cr * cr = eight THEN
FOR v = 0 TO 9
IF used(v) = 0 THEN
used(v) = 1
FOR n = 0 TO 9
IF used(n) = 0 THEN
used(n) = 1
seven = (((s * 10 + e) * 10 + v) * 10 + e) * 10 + n
IF seven MOD 2 > 0 AND seven MOD 3 > 0 AND seven MOD 5 > 0 AND seven MOD 7 > 0 THEN
IF seven MOD 11 > 0 AND seven MOD 13 > 0 AND seven MOD 17 > 0 AND seven MOD 19 > 0 THEN
PRINT six, seven, eight
END IF
END IF
used(n) = 0
END IF
NEXT n
used(v) = 0
END IF
NEXT v
END IF
used(t) = 0
END IF
NEXT t
used(h) = 0
END IF
NEXT h
used(g) = 0
END IF
NEXT g
used(e) = 0
END IF
NEXT e
END IF
used(x) = 0
END IF
NEXT x
used(i) = 0
END IF
NEXT i
used(s) = 0
NEXT s

The results are:

`380           36467         68921420           43139         32768420           43531         32768420           43931         32768930           91517         13824`

In only the first instance is the middle number, SEVEN, prime, as

43139=179*241
43531=101*431
43931=197*223
91517=23*23*173

 Posted by Charlie on 2005-10-05 14:42:32

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