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Six Seven Eight (Posted on 2005-10-05) Difficulty: 2 of 5
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    
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Solution 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         68921
420           43139         32768
420           43531         32768
420           43931         32768
930           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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