735 is a 3-digit base ten positive integer formed by a permutation of the digits 3, 5 and 7 and, 735 is divisible by each of 3, 5 and 7.
Determine the minimum value of N with N ≥ 11 such that there exists a 3-digit base-N positive integer p which is formed by a permutation of the digits 3, 5 and 7 such that p is divisible by each of 3, 5 and 7. What is the next smallest value of N with this property?
DECLARE SUB factor (num#, s$)
DECLARE SUB permute (a$)
DEFDBL A-Z
CLS
FOR n = 11 TO 500
s$ = "357"
h$ = s$
DO
v = VAL(LEFT$(s$, 1)) * n * n + VAL(MID$(s$, 2, 1)) * n + VAL(RIGHT$(s$, 1))
IF v MOD 3 = 0 AND v MOD 5 = 0 AND v MOD 7 = 0 THEN
factor v, fct$
PRINT n, s$, v, fct$
ct = ct + 1
END IF
permute s$
LOOP UNTIL s$ = h$
IF ct >= 45 THEN EXIT FOR
NEXT n
SUB factor (num, s$)
s$ = "": n = ABS(num): IF n > 0 THEN limit = SQR(n): ELSE limit = 0
IF limit <> INT(limit) THEN limit = INT(limit + 1)
dv = 2: GOSUB DivideIt
dv = 3: GOSUB DivideIt
dv = 5: GOSUB DivideIt
dv = 7
DO UNTIL dv > limit
GOSUB DivideIt: dv = dv + 4 '11
GOSUB DivideIt: dv = dv + 2 '13
GOSUB DivideIt: dv = dv + 4 '17
GOSUB DivideIt: dv = dv + 2 '19
GOSUB DivideIt: dv = dv + 4 '23
GOSUB DivideIt: dv = dv + 6 '29
GOSUB DivideIt: dv = dv + 2 '31
GOSUB DivideIt: dv = dv + 6 '37
IF INKEY$ = CHR$(27) THEN s$ = CHR$(27): EXIT SUB
LOOP
IF n > 1 THEN s$ = s$ + STR$(n)
EXIT SUB
DivideIt:
DO
q = INT(n / dv)
IF q * dv = n AND n > 0 THEN
n = q: s$ = s$ + STR$(dv): IF n > 0 THEN limit = SQR(n): ELSE limit = 0
IF limit <> INT(limit) THEN limit = INT(limit + 1)
ELSE
EXIT DO
END IF
LOOP
RETURN
END SUB
finds
base representation decimal factors
equivalent
19 753 2625 3 5 5 5 7
26 537 3465 3 3 5 7 11
31 357 3045 3 5 7 29
31 735 6825 3 5 5 7 13
49 357 7455 3 5 7 71
54 753 20685 3 5 7 197
56 537 15855 3 5 7 151
61 537 18795 3 5 7 179
61 753 26355 3 5 7 251
80 735 45045 3 3 5 7 11 13
91 357 25305 3 5 7 241
91 537 41685 3 5 7 397
94 357 26985 3 5 7 257
96 753 64995 3 5 7 619
101 735 71715 3 5 7 683
115 735 92925 3 3 5 5 7 59
124 753 108255 3 5 7 1031
131 537 86205 3 5 7 821
136 357 56175 3 5 5 7 107
136 735 129885 3 5 7 1237
154 357 71925 3 5 5 7 137
159 753 177765 3 5 7 1693
161 537 130095 3 3 5 7 7 59
166 537 138285 3 3 5 7 439
166 753 193725 3 3 3 5 5 7 41
185 735 240135 3 5 7 2287
196 357 116235 3 3 3 3 5 7 41
196 537 192675 3 5 5 7 367
199 357 119805 3 5 7 7 163
201 753 283815 3 3 5 7 17 53
206 735 297675 3 3 3 3 3 5 5 7 7
220 735 339465 3 5 7 53 61
229 753 368235 3 3 5 7 7 167
236 537 279195 3 5 7 2659
241 357 175455 3 3 5 7 557
241 735 407295 3 3 3 5 7 431
259 357 202545 3 3 5 7 643
264 753 489195 3 3 5 7 1553
266 537 354585 3 5 7 11 307
271 537 368025 3 5 5 7 701
271 753 515445 3 5 7 4909
290 735 589575 3 5 5 7 1123
301 357 273315 3 5 7 19 137
301 537 453915 3 3 5 7 11 131
304 357 278775 3 3 3 5 5 7 59
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Posted by Charlie
on 2013-03-03 12:17:56 |
computer solution
