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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