Sets and Numbers

Probability Level 4

Let A denote a subset of the set { 1,11,21........541,551} having the property that no two elements of A add up to 552. Lets assume that "N" is the maximum number of elements A can have. Find the number of zeroes in the product of the first "N" numbers then.


The answer is 6.

Krishna Ar by Krishna Ar , 21, India

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1 solution

My Goodness! Is this a combinatorics problem or number theory?

OK, Here goes the solution:

First, we can figure out that the given set has 56 elements

Let us try to make the required subset.

Note that for every element we select, there is an element we reject because it would otherwise make a 552 pair

For example, if we choose 1, we reject 551.

Thus, we can choose atmost 56/2 or 28 numbers.

The first 28 numbers in the set are {1, 11, 21, 31, 41, 51, 61, 71, 81, 91, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 201, 211, 221, 231, 241, 251, 261, 271}, the product of which is 15946179081353016930375083310371967357185112316115801481

and I have not the slightest idea of evaluating it without a computer

2 Helpful 1 Interesting 0 Brilliant 0 Confused

Very good solution

swapnil rajawat - 7 years ago

That's a very interesting evaluation of the problem, I thought that the question asked was quite different from what you thought it was asking, but both ways produce the same answer in the end :)

I believe that the question is asking for the number of zeroes in the product of the first N numbers, N being defined as the maximum number of elements A can have. N is defined as a number, and not a set. Hence, the question is asking for N! where N is the number of elements, not the numbers in the set. Since we can choose a maximum of 28 numbers for the set, N is 28, and we can find the number of zeroes in 28!

Zeroes can be found simply by how many 5's there are in the prime factorization of 28!

There are 5's in 5, 10, 15, 20, and there are two in 25, so the answer is 6 zeroes in the product of the first N numbers.

Byungjun Kim - 6 years, 11 months ago

what software do you use to calculate such results ?

Apoorv Padghan - 6 years, 10 months ago

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