I need help as soon as possible with the problem bellowSimplify each of the following with justifications for your answer:
a. 290 mod 26
b. 9^{-1} mod 26
c. -15 mod 26
Answers
We subtract the largest multiple of 26 which is less than 290. The answer we get after the subtraction is the answer.
Largest multiple of 26 is 286, because 26 * 11 = 286
290 - 286 = 4.
290 mod 26 = 4.
b). 9^(-1) mod 26 = 0.111 mod 26.
0 * 26 = 0.
The largest multiple of 26 which is less than 0.111 is 0.
We subtract this from 0.111
0.111 - 0 = 0.111
9^(-1) mod 26 = 9^(-1) = 0.111
c). -15 mod 26.
Take note this is negative numbers. But same rules applies here.
The largest multiple of 26 which is less than -15 is -26.
We subtract this from -15
-15 - (-26) = -15 + 26 = 11
-15 mod 26 = 11
Final answer:
The results of the modulo operations are: 290 mod 26 results in 2; 9^{-1} mod 26 is equivalent to 3; and -15 mod 26 equals 11.
Explanation:
To simplify each of the following with justifications:
- 290 mod 26: When you divide 290 by 26, the remainder is 2. Therefore, 290 mod 26 = 2.
- 9^{-1} mod 26: First, you need to find the multiplicative inverse of 9 in mod 26. The multiplicative inverse of an integer a modulo m is an integer b such that (ab) mod m = 1. For 9, the multiplicative inverse is 3, since (9*3) mod 26 = 27 mod 26 = 1. Therefore, 9^{-1} mod 26 = 3.
- -15 mod 26: In modular arithmetic, if you end up with a negative number, you add the modulus until you get a positive result. Here, adding 26 once to -15 gives you 11. Therefore, "-15 mod 26 = 11".
Learn more about Modulo Operations here:
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