SATYAJIT Ray was not only a good film director, but also a talented artist and mathematician. The incident of Newton Dutt could easily have happened on the sets of one of his films - mathematics and art are so intertwined. An Oscar for lifetime achievement in cinema is just one aspect of his life that makes him so dear to us. The story of Newton Dutt shows that even persons whom we take to be mathematically challenged can surprise us with their logic and that mathematical discoveries can be made anytime, anywhere and by anyone.

I have three best explanations of Newton's answer from three regular readers. Please forgive me, I can't stop bugging you; don't decry me, it's short-cut you are going through …ooo…please forgive me… - everything I do, I do it for you. The problem was: What is the smallest integer greater than 0 that can be written entirely with zeros and ones and is evenly divisible by 225? The solution Newton gave was 11111111100 and you had to tell why.

The prime factorisation of 225 is 5*5*3*3; so, the answer will be both a factor of 25 and of 9. All factors of 25 end in 00, 25, 50, or 75. The only one of these composed of 0s and 1s is obviously 00, so, the answer must end in 00. The difficult thing is finding a series of 0s and 1s before the 00 that will make the entire number divisible by 9. The trick is that if the sum of digits of a number is divisible by 9, then, the number is also divisible by 9. For example, the number 17,685 is divisible by 9 because 1+7+6+8+5=27, and 27 is divisible by 9.

To prove this let's consider any five-digit number abcde, which can be expressed as follows: a*10000 + b*1000 + c*100 + d*10 + e =

a*(9999+1) + b*(999+1) + c*(99+1) + d*(9+1) + e*1 =

a*9999 + b*999 + c*99 + d*9 + a + b + c + d + e =

9*(a*1111 + b*111 + c*11 + d*1) + a + b + c + d + e;

9*(a*1111 + b*111 + c*11 + d*1) must be divisible by 9 because it is a factor of 9, so, if a+b+c+d+e is also a factor of 9, then, the entire number must be a factor of 9. The remainder of abcde/9 is the same as the remainder of (a+b+c+d+e)/9. The smallest number consisting of all 1s and divisible by 9 is, thus, 111,111,111. Adding the two zeros at the end gives the answer to the problem: 11,111,111,100.

The three who gave the best solutions are Shubhangi of Urban Estate in Kurukshetra, Rohit Pardasani of Sector 29-D in Chandigarh and Ravinder Mittal; and their answers are based on the logic given here only, with deviations of style. The logic here gives a sort of general rule for solving such problems.

You can find several instances of mathematics in movies, but don't go too far in this pursuit. A boy, once, got a beating from his parents because he told them he was watching 'The Pianist' to prepare for the mathematics test that was on the next day. 'The Pianist' is based on the life of a famous scientist and includes interesting mathematical problems, but the parents didn't know. Don't face such music, do well in examinations and keep writing at The Tribune or adityarishi@yahoo.co.in.