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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?

 Sep 14, 2022
 #1
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1) \({19 \over t} + {5 \over t} = {24 \over t}\)

 

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

 Sep 14, 2022
 #3
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So it would be all the divisors of 24, which is 1, 24, 2, 12, 3, 8, 4, and 6, so the answer would be 8?

Bloom  Sep 14, 2022
edited by Bloom  Sep 14, 2022
 #6
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Basically, t is all the divisors of 24. 

 

There is more than 1 answer.

BuilderBoi  Sep 14, 2022
 #2
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2) \(1989 = 3^2 \times 13 \times 17 \), so \(\color{brown}\boxed{3}\) prime divisors. 

 Sep 14, 2022
 #4
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Thank you!

Bloom  Sep 14, 2022
 #5
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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 \(2009 + 7 - 1 = \color{brown}\boxed{2015}\) 

 Sep 14, 2022

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