SOLUTION: Can you please help me? Can you give me answers to those problems and explain how to solve them? Thanks a lot!!!! 1. The inverse of a modulo 39 is b. What is the inverse of 4a mod

Question 486215: Can you please help me? Can you give me answers to those problems and explain how to solve them? Thanks a lot!!!!
1. The inverse of a modulo 39 is b. What is the inverse of 4a modulo 39 in terms of b?
2. Let x and y be integers satisfying 41x + 53y = 12. Find the residue of x modulo 53.
3. John and Ling start their new jobs on the same day. John's schedule is 4 workdays followed by 1 day off. Ling's schedule is 7 workdays followed by 2 days off. On how many days during their first year of work (365 days) do John and Ling have the same day off?
Thank you very much!!!
Answer by chessace(471) (Show Source): You can put this solution on YOUR website!
1. b = inverse of (variable) a mod 39 means a + b = 0 mod 39.
As usual let x = answer = inverse 4a mod 39.
So x + 4a = 0 mod 39.
Substract 4a+4b, which is legal because 4(a+b) = 4*0 = 0 all mod 39
Result: x - 4b = 0 mod 39. or
x = 4b mod 39.
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2. 41x + 53y = 12, find x mod 53.
53 mod 53 is 0, so the y term goes away.
41 + 12 = 53 = 0 mod 53, so 41x = -12x mod 53
-12x = 12 mod 53
x = -1 mod 53
Normalize to residue: x = 52 mod 53
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3. The mods to use are the total cycle = work + off.
J's days off are all those = 0 mod 5 from start 1.
L's days off are all those = 0 or 8 mod 9 from start 1.
So every 5*9 = 45 day interval looks the same to them.
Within that interval, J is off 9 times which are days
(mod 9) 5 1 6 2 7 3 8 4 0.
You could leave out this step knowing all digits will appear once, but there is a fraction in the works, so order matters.
Two days match every cycle, 365/45 = 8.11 cycles (no match in .11)
So 16 days off in common that 1st year.

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