Question 1205483
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Let A = {{{(b*(b-1)*(b-2)*ellipsis*(1))/(b+1))}}}, where b is an integer and 1 < b < 60. 
For how many values of b is A an integer? 
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<pre>
If  b+1  is a prime number, then obviously

     {{{(b*(b-1)*(b-2)*ellipsis*(1))/(b+1))}}}

is not an integer number.


If  b+1  is NOT a prime number, then  b+1 = p*q,  where p and q are integer numbers lesser than (b+1).

Hence, p and q are among the numbers in the numerator, and they can be canceled,
producing integer ratio.


Thus, the given expression is integer if and only if (b+1) is not a prime.


So, to answer the question, we need to count the number of primes in the interval 1 < b < 60.


These primes are  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,  
and their number is 17.



At this point, we obtain the  <U>ANSWER</U>: the number of integers b, 1 < b < 60, such that      

    {{{(b*(b-1)*(b-2)*ellipsis*(1))/(b+1))}}}  is integer 

is equal to (59-1)-17 = 58-17 = 41.


In this reasoning, there is only one weak point: what if in the decomposition b+1 = p*q
the integers p and q are equal ?


Then one of them cancels, and for the other equal factor in the denominator, there is still another 
number in the numerator, which is multiple to it, so in this case we AGAIN obtain an integer number. 
</pre>

Solved. The proof is complete.