a) How many positive integers t will make the expression 19/t + 5/t have an integral value?

b) How many different prime divisors does 1989 have?

c) If a year had 364 days, then the same calendar could be used every year by only changing the year. A "regular" year has 365 days and a leap year has 366 days. The year 2000 was a leap year and leap years occur every 4 years between the years 2000 and 2100. Claudia has a calendar for 2009. What will be the next year that she can use this calendar by merely changing the year?

Bloom Sep 14, 2022

#1**+1 **

1) \({19 \over t} + {5 \over t} = {24 \over t}\).

For t to work, it must be a divisor of 24.

BuilderBoi Sep 14, 2022

#2**+1 **

2) \(1989 = 3^2 \times 13 \times 17 \), so \(\color{brown}\boxed{3}\) prime divisors.

BuilderBoi Sep 14, 2022

#5**+1 **

3) Note that because a year has 1 exta day, you will need to wait 7 years.

But, on 2012 (leap year), we have 2 extra days, so it is \(2009 + 7 - 1 = \color{brown}\boxed{2015}\)

BuilderBoi Sep 14, 2022