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       Proposed Problems

Proposed by   John Morse Delmar NY    Tel: 518-439-3543

 

   1.  How many pairs of prime numbers are there such that their
       sum is forty?           

   2.  What is the only prime number between 200 and 220?      

   3.  The sides of triangles A and B measure 5, 5, 8 and 5, 5, 6
       respectively.  What is the ratio of the area of triangle A
       to that of triangle B?  Express in simplest a:b form.      

   4.  Let P be the product of any group of 2-digit prime numbers.
       Which digits CANNOT be in the ten's place of five times P?    

   5.  Given the fraction N/210, what is the LEAST value of positive
       integer N such that:  a) N is composite, and   b) N/210 is
       already in lowest terms?   (1 is neither composite nor prime.)

   6.  If a perfect square ends with the non-zero digits NNN, then
       which digit is N?          

   7.  If A/B + 4/3 + 9/2 = A/B x 4/3 x 9/2, then find the value of
       A/B in lowest terms; express in A/B format.       

   8.  Let K be an integer greater than 4 such that K+2, K+4, K+8,
       and K+10 are all prime. What is the greatest integer that
       is ALWAYS a factor of K+6?         

   9.  Let M be the least common multiple of all the consecutive
       integers 10 through 30, inclusive.  Let N be the least
       common multiple of M, 32, 33, 34, 35, 36, 38, 39, and 40.
       What is the value of N divided by M?        

  10.  For any positive integer G, what is the greatest common
       divisor of 2G+1 and 2G+5?           

  11.  What is the least positive integer N such that 450N is a
       perfect cube?           

  12.  Let S1 be the set of all positive multiples of 2 that are less
       than 101.  Then, let S2 be the set of all positive multiples
       of 3 that are also less than 101.  Finally, let S3 be the set
       of all positive multiples of 5 that are less than 101.  What
       is the sum of all the numbers occurring in all three sets?    

  13.  What is the least multiple of 14 whose digital sum is 14?     

  14.  144, being a multiple of itself, naturally ends with ...144.
       What is the next greater multiple of 144 ending in ...144?    

  15.  What is the only ordered triple {a,b,c} such that 6a+9b+20c=61,
       where a, b, and c are positive whole numbers only?      

  16.  x, y, and z are three consecutive whole numbers whose sum is
       odd.  Let P be the product of x, y, and z, and let M be the
       least common multiple of x, y, and z.  What is the ratio of
       M to P? Express in simplest a/b form.        

  17.  Express 9991 as the product of its prime factors.      

  18.  Let abc be a 3-digit whole number, where a, b, and c are
       its digits, not necessarily different.  If nine times abc
       is 1abc, what number is abc?         
       

       SOLUTIONS

       1)  3      2)  211    3)  1:1  4)  2 & 7     5)  121

       6)  4      7)  7/6    8)  15  9)  2       10)  1

      11)  60     12)  180   13)  266 14)  18144    15)  {2,1,2}

      16)  1/2     17)  97x103   18)  125

"Chance favors the prepared mind." - Louis Pasteur