:) three dogs barked all night last night. lola, patch, and lady began barking together at 11: 00 p. m. then, lola barked every 5 minutes, patch every 8 minutes, and lady every 12 minutes. later that night, ms. li was awakened when all 3 dogs barked together again. at what time was ms. li awakened? show your work.

1:00 a.m. (and 3:00 a.m., and 5:00 a.m.)

The problem is to find the least common multiple (LCM) of the barking intervals. One way to find the LCM is to find the product of the factors of the numbers.

... 5 = 5 . . . . a prime number

... 8 = 2³

... 12 = 2²×3

The factor 2³ is a multiple of the factor 2², so we only need 2³ in our final product.

The LCM is then ...

... 2³×3×5 = 120 . . . . minutes

The dogs will all bark together every 120 minutes (2 hours).

2 hours after 11 p.m. is 1 a.m.

The problem statement doesn't say whether Ms. Li was awakened multiple times. The dogs will bark together again at 3 a.m., 5 a.m. and (possibly) 7 a.m.

probability for j is 1/4

while

a is 1/6