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One More Factor, Please (Posted on 2008-03-23) Difficulty: 3 of 5
Four three-digit numbers are in arithmetic progression, and the number of factors of each is also in arithmetic progression. In fact that second arithmetic progression has a constant difference of 1, so that each successive number in the original arithmetic progression has one more factor than the number before it.

Note: These are not the prime factors of the numbers, but rather any factor, including the number itself and 1, so, for example, 46 has four factors (1, 2, 23 and 46), as does 8 (1, 2, 4 and 8).

What is the original arithmetic progression of three-digit numbers?

  Submitted by Charlie    
Rating: 3.3333 (3 votes)
Solution: (Hide)
number          factors
 229           2  (1, 229)
 361           3  (1, 19, 361)
 493           4  (1, 17, 29, 493)
 625           5  (1, 5, 25, 125, 625)

The numbers being factored differ by 132 from one to the next.

DECLARE FUNCTION factors# (n#)
DEFDBL A-Z

CLS
DIM nfact(100 TO 999)

FOR n = 100 TO 999
 nfact(n) = factors(n)
NEXT

FOR strt = 100 TO 999 - 3
  FOR diff = 1 TO 333
    IF strt + 3 * diff > 999 THEN EXIT FOR
    dif2 = nfact(strt + diff) - nfact(strt)
    IF dif2 = 1 THEN
      IF nfact(strt + 2 * diff) = nfact(strt + diff) + dif2 THEN
        IF nfact(strt + 3 * diff) = nfact(strt + 2 * diff) + dif2 THEN
         FOR i = strt TO strt + diff * 3 STEP diff
          PRINT i, nfact(i)
         NEXT
         PRINT : PRINT diff
        END IF
      END IF
    END IF
  NEXT
NEXT

FUNCTION factors (n)
 test = n: lim = INT(SQR(n) + .5)
 ct = 0
 FOR i = 1 TO lim
   q = INT(test / i + .5)
   IF q * i = test THEN
     IF q = i THEN ct = ct + 1:  ELSE ct = ct + 2
   END IF
 NEXT
 factors = ct
END FUNCTION

Based on Enigma No. 1479 by Richard England, New Scientist, 2 February 2008.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
AnswerK Sengupta2008-12-31 00:39:16
Solutionre: Solution / method ==>>MY WAYAdy TZIDON2008-03-24 20:43:13
Solution / methoded bottemiller2008-03-24 15:19:59
SolutionSolutionDej Mar2008-03-23 21:34:02
one solutionxdog2008-03-23 12:51:04
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