Maths-
General
Easy

Question

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is r squared s to the power of 4 t squared, then the number of ordered pair (p, q) is-

  1. 224    
  2. 225    
  3. 252S    
  4. 256    

hintHint:

At least one of the p and q, must have r,s and t in their prime factorization.
Now, we have to consider each case for selecting the number of ways for power of rs and t . So, in this way we have to approach the problems.

The correct answer is: 225


    Step by step solution :
    It is given that L C M space left parenthesis p comma q right parenthesis space equals space r squared s to the power of 4 t squared
    That is, at least one of p and q must have r squareds to the power of 4 and t squared in their prime factorizations.
    Now, consider the cases for power of r as follows:
    Case 1 : p contains  s to the power of 4 then q has  s to the power of K with K = (0,1,2,3)
    That is, number of ways=4.
    Case 2 : q contains  s to the power of 4 then q has  s to the power of K with K = (0,1,2,3)
    That is, number of ways=4.
    Case 3 : Both p and q contains s to the power of 4 
    Then, number of ways=1.
    Therefore, exponent of r may be chosen in 2+2+1=5 ways.
    Similarly, exponent of s may be chosen in 4+4+1=9 ways and 
    Exponent of s may be chosen in 2+2+1=5 ways
    Thus, the total number of ways is:
     5 space cross times 9 space cross times 5 space equals space 225
    Hence, the number of the ordered pair (pq) is 225.

    Be careful about selecting the r s and t for this particular problem. Because number ways for each selection of r, s and t there are 3 cases. Because there are two values given

    p, q . Hence to calculate the total number of ways is to multiply the number of ways for r, number of ways for s and number of ways for t. So, in this way to solve this particular type of problem.

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