Category Archives: brain puzzler

October 2019 Brain Puzzler Solution

Q:  How many factors of 155^9 are perfect squares and/or perfect cubes?

A: 37 factors of 155^9 are perfect squares and/or perfect cubes.

155 is a composite number composed of 5 x 31.
Therefore
155^9 = 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 31 x 31 x 31 x 31 x 31 x 31 x 31 x 31 x 31

25 Perfect Squares:
1: 1 is both a perfect square and a perfect cube.
4:  Perfect squares with just 5s = 5^2, 5^4, 5^6, 5^8
4:  Perfect squares with just 31s = 31^2, 31^4, 31^6, 31^8

Combining the 5s and the 31s Perfect Squares
4: 5^2 x 31^ 2, 5^4 x 31^2, 5^6 x 31^2, 5^8 x 31^2 
4: 5^2 x 31^ 4, 5^4 x 31^4, 5^6 x 31^4, 5^8 x 31^4 
4: 5^2 x 31^ 6, 5^4 x 31^6, 5^6 x 31^6, 5^8 x 31^6 
4: 5^2 x 31^8, 5^4 x 31^8, 5^6 x 31^8, 5^8 x 31^8 

12 Perfect Cubes:
Perfect cubes with just 5s = 5^3, 5^6 (already counted above), 5^9  (2)
Perfect cubes with just 31s = 31^3, 31^6 (already counted above), 31^9 (2)

Combining the 5s Perfect Cubes and the 31s Perfect Cubes
5^3 x 31^3, 5^6 x 31^3, 5^9 x 31^3  (3)
5^3 x 31^6, 5^9 x 31^6 (2)
5^3 x 31^9, 5^6 x 31^9, 5^9 x 31^9  (3)

August 2019 Brain Puzzler Solution

Q: On an analog clock, how many times do a clock’s hands overlap in a one week period?

A: 154
At 12:00, the hands exactly overlap but not at other times — for example, at 1:05, the hour hand has moved slightly… and at 6:30, the hour hand is halfway between the 6 and the 7.

Making the times that they will overlap only 11 times in 12 hours (60 minutes /11).

12:00, 1:05ish, 2:10ish, 3:15ish, 4:20ish, 5:25ish, 6:30ish, 7:35ish, 8:40ish, 9:45ish, 10:50ish.
In one day they will overlap 22 times so in one week, 154 times.