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Counting triplets II (Posted on 2011-07-30) Difficulty: 3 of 5
Each of A, B and C is a positive integer with A < B < C < 300

Determine the total number of triplets (A, B, C) such that;

(I) A+B+C is divisible by 3, and:

(II) sod(A)*sod(B)*sod(C) is divisible by 5.

Bonus Questions:

(A) Disregarding condition (I), determine the total number of triplets.

(B) Disregarding condition (II), determine the total number of triplets.

Note: sod(x) refers to the sum of digits of x.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution Comment 1 of 1

DECLARE FUNCTION sod# (x#)
DEFDBL A-Z
CLS
FOR a = 1 TO 297
FOR b = a + 1 TO 298
FOR c = b + 1 TO 299
  c1fl = 0: c2fl = 0
  IF (a + b + c) MOD 3 = 0 THEN
    c1ct = c1ct + 1
    c1fl = 1
  END IF
  IF (sod(a) * sod(b) * sod(c)) MOD 5 = 0 THEN
    c2ct = c2ct + 1
    c2fl = 1
  END IF
  IF c1fl AND c2fl THEN bothct = bothct + 1
NEXT
NEXT
NEXT

PRINT c1ct; c2ct, bothct

FUNCTION sod (x)
  s = 0
  st$ = LTRIM$(STR$(x))
  FOR i = 1 TO LEN(st$)
    s = s + VAL(MID$(st$, i, 1))
  NEXT
  sod = s
END FUNCTION

finds

1470249  2135269            712009

meaning 712,009 meet both criteria and:

1,470,249 meet the first criterion and this constitutes the answer to B.

2,135,269 meet the second criterion and this is then the answer to part A.

Those that meet both criteria are also counted in the individual criterion counts.

These are out of the 4,410,549 possible combinations of numbers.


  Posted by Charlie on 2011-07-30 17:42:18
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