Question 1082072: Let ⊕ be the operation of addition modulo 26 and ⊙ be the operation of multiplication modulo 26 defined on Z26. Perform the indicated operations:
a) 22 ⊕ 19 ; Z26=15
b) 15 ⊙ 7 ; Z26=1
c) 25 ⊕ 9 ⊙ 6 ; Z26=1
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! You have the answers:
a) In regular arithmetic,  ,
so the result in modulo 26 is  .
b) In regular arithmetic,  ,
so the result in modulo 26 is  .
c) Order of operation convention, extended to these two new operations,
dictates that 25 ⊕ 9 ⊙ 6 = 25 ⊕ ( 9 ⊙ 6 )
In regular arithmetic,  ,
so the 9 ⊙ 6 term is 2 in modulo 26.
Then, you have to calculate 25 ⊕ 2 .
In regular arithmetic,  , so
25 ⊕ ( 9 ⊙ 6 ) = 25 ⊕ 2 = 1 .
NOTE:
Modular arithmetic is something you use daily,
but in math class, it involves a bunch of symbols and jargon
that unfortunately you may be expected to remember.
In modular arithmetic (which some would call clock arithmetic),
you calculate results as you do in your daily life with time.
ours in an analog clock are in a circle, and after 12, you get back to 1.
If it is 8 AM, you know that in 5 hours it will be 1 PM,
because 8+5=13, but AM/PM time only goes up to 12,
and 13 is 1 more than 12 (13-12=1).
You do those modulo 12 calculations before learning about modular arithmetic in math class.
For military time (where 1PM is 1300), or 24-hour time (where 1PM is 13:00),
you would use modulo 24.
You add up the hours; divide the result by 24;
take the result as days passed,
and take the remainder as time that day.
If the time is 8:00 PM, (2000 in military time),
and you get a 34-hour leave,
you know it ends at 6:00 AM (0600) the second day after today,
because 2000+3400=5400,
and when dividing 5400 by 2400 (or 54 by 24),
you get 2 with a remainder of 6.
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