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Year from Ratio (Posted on 2016-08-22) Difficulty: 3 of 5
S denotes the sum of the digits and P denotes the product of the nonzero digits.
It is observed that for the year 2016, S = 9 and P = 12, so that:
S:P = 3:4

Determine all years from 2000 to 3000 such that:
S:P = 3:4

No Solution Yet Submitted by K Sengupta    
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Solution No Subject | Comment 2 of 3 |

We are looking      for a triplet ABC in  number 2ABC, denoting the year in question.
After finding the triplet we will list all possible permutations (6,3, or 1).

Wlog  we assume  A being the lowest digit  and A<3, 2*3*3*3=54  &  (3/4*54- 2)> 9+9+9.

The candidate years begin:  20**,21**,22**
(+ permutations)

Trying  A=0, 1 , 2    in:

(2+A+B+C)/2ABC=3/4 OR

2(2+A+B+C)=3*A*B*C

 We get:

20:     2004, 2022, 2040, 2106, 2160, 2202, 2220, 2400, 2601, 2610

21:   2118, 2181, 2811.

22: no ans.

 

Edited on August 23, 2016, 10:45 am
  Posted by Ady TZIDON on 2016-08-23 10:39:44

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