Need help solving problem Solving! Explanation plus answer would be great!

I want to tell you about Elmo and the problem he was hiving with nightmares. Elmo was having some bad recurring nightmares. Every 19 days he was having a nightmare about 19 ghosts scaring him our of his wits by jumping out of a 1919 Hupmobile and saying "boo" 19 times. He was also having a nightmare every 13 days about 13 black cats crossing his path in from of his house, located at 1313 Thirteenth Street. He had a bad week in April when on the night of Friday, April 5, he had the nightmare about the 19 ghosts saying "boo" 19 times. The very next night he had the nightmare about the 13 black cats. Elmo know he can handle these nightmares when they occur from time to time, but lately he's been really worried about them both occurring on the same night. Do they ever occur on the same night by the end of the year, and if so, on what date?!?!?

First, get rid of the extra information.
This says: Event X happens every 19 days and it happened on April 5th. Event Y happens every 13 days and it happened on April 6th. Will the events occur on the same day by Dec 31st?


Let's say they do occur on the same day, and let's give it a name: suppose they occur Z days after April 5th (so, for example, if Z= 2, it would occur on April 7th)

Then Z should be divisible by 19, because Event X will have happened n times (If that's confusing, think about it this way: if he dreams about ghosts once more, 19 days will have passed. If he dreams about ghosts twice more, 2*19 days will have passed. 10 more times, 190 days will have passed)

So we know that Z=19*n, where n is an integer.

We also know that Z -1 = 13*m, because Z days after Apr. 5th is Z-1 days after April 6th, so Z-1 must be divisible by 13.

So now we're looking for Z so that Z=19n and Z= 13m +1, or we're looking for n and m so that

19n = 13m+1

Just list the multiples of 19:

19
38
57
76
95
114
133
152
171
190
209
228
247
285
304

Now, subtract 1 from all of those and determine which of those will be divisible by 13. You'll see that
209 =11*19 = 1 + 16*13, so both X and Y will happen 209 days after April 5th. The last thing we need to determine is what day that is.

April 5th is the 95th day of the year (assuming it's not a leap year) , so we're looking for the 304th day of the year, which turns out to be Oct 31st, or halloween.

Cat of any colour crossing in front when we go out in car or walk is very bade sign. It will indicate the purpose you are going out will not be in good ending
If the cat crossing from Left to wright it is very dangerous (increasing order of Alphabet( L,M,N, N---W) .
But if it is in desending order , means crossing wright sise to left side, you are safe.
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