help please?!

Question

Suppose we call a number funny if it is the product of three prime factors, of which exactly two are the same. (For example: 12 = 2 x 2 x 3, so 12 is funny.) What is the total number of funny numbers between 30 and 60?

Answer 1

Debate Topic:
Resolved:  Questions that cause you to smile are characteristically better than questions that don't.
 
This is a cool question!  And, it has several cool solutions (at different levels of math).
 
Now, on the pretty complicated side, you could study the products of prime numbers so that you could decrypt coded messages between computers (see "RSA problem" on Wikipedia) or, on the simple side, just to find an LCM (least common multiple) of some numbers.
 
Here is a cool solution:
  The only prime factors we must consider are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.  That's because multiplying any bigger number by 2 (the very smallest prime) gives a result greater than 60.
  Now, if the number is to be funny, it must have at least two of the same prime numbers as a factor (group them together), so a grouped factor of the number must be:
                                                         4, 9, 25     (anything larger times 2 also exceeds 60)
O.K., now which of these 3 values can be multiplied by a single prime number to get a result between 30 and 60? (just try them -- iterate with a number in the second list and a number in the first list):
         4 *11 = 44
         4 *13 = 52

         9 * 5 = 45

 

         25 * 2 = 50
 
Looks like the funny numbers between 30 and 60 are 44, 52, 45, and 50.    (note: I left them out of order)

Answer 2

 

2*2*11=44

2*2*13=52

3*3*5=45

5*5*2=50

there are 4 funny numbers

or...

31

32=2*2*2*2*2

33=3*11

34=2*17

35=5*7

36=2*2*3*3

37

38=2*19

39=3*13

40=2*2*2*5

41

42=2*3*7

43

44=2*2*11

45=3*3*5

46=2*23

47

48=2*2*2*2*3

49=7*7

50=2*5*5

51=3*17

52=2*2*13

53

54=2*3*3*3

55=5*11

56=2*2*2*7

57=3*19

58=2*29

59