Download Notes on Modular Arithmetic: Definitions, Rules, and Applications and more Study notes Computer Science in PDF only on Docsity! Notes about mod Arup Guha First we’ll define divisibility. We say that a | b if and only if there is some integer c such that b = ac. In English, “a | b” would be read as “b is divisible by a.” For example, 6 | 18, 197 | 0 and 34 | 34. Now, let’s define mod: a ≡ b (mod n) if and only if n | (a – b). (This just means there exists some integer c such that a – b = nc.) In essence, this is true if n divides evenly into the difference of a and b. Alternatively, we can think of it as follows: when a and b are divided by n, they leave the same remainder. In our class, typically we will make some mathematical calculation and then we’d like to know what letter a particular number corresponds to. What we really want is give some integer a, we want to find a value b such that 0 ≤ b < 26 and a ≡ b (mod 26). For example, if we get 194 after some calculation and want to know what letter it is, our goal is to find the unique value of b such that 194 ≡ b (mod 26), with 0 ≤ b < 26 We can determine that 194 ≡ 12 (mod 26). We can verify this because 194 – 12 = 182 and 182 = 26x7. The easy way to find b when the starting value is greater than 26 is to divide 26 into the number. When we divide 26 into 194, it goes in 7 times, leaving a remainder of 12, which is our desired value. Consider a second example: -85 ≡ b (mod 26), with 0 ≤ b < 26 By dividing, we find that -85 ≡ -7 (mod 26), since -85 – (-7) = -78 and -78 = 26x(-3), but we also see that we haven’t gotten the desired value of b either. We can simply add 26 to -7 to do that, since adding or subtracting multiples of 26 will “create” other values equivalent to the original. Thus, we have: -85 ≡ -7 ≡ 26 – 7 ≡ 19 (mod 26) Now, let’s look at some rules with mod: if a ≡ b (mod n), then a + c ≡ b + c (mod n) if a ≡ b (mod n), then ac ≡ bc (mod n) and ac ≡ bc (mod cn), but this latter fact is rarely used if a ≡ b (mod n), then ak ≡ bk (mod n) if a ≡ b (mod n) and c ≡ d (mod n), then a+c ≡ b+d (mod n), and ac ≡ bd (mod n) These are fairly straight-forward to apply. However, division rules are tricky since we are now dealing with integers. If we have a situation such as 3a ≡ 16 (mod 26) we deal with it by multiplying through by the inverse of 3 (mod 26) which is 9, to yield the following equation: 9(3a) ≡ 9(16) (mod 26) 27a ≡ 144 (mod 26) a ≡ 14 (mod 26) Here is a list of the inverses mod 26: 1 3, 9 5, 21 7, 15 11, 19 17, 23 25 (Note: 1 is an inverse of itself as is 25. The rest are pairs, so 3 is the inverse of 9 and 9 is the inverse of 3 (mod 26), etc.) But what about an equation like 4a ≡ 14 (mod 26) or 4a ≡ 7 (mod 26) This literally means: 4a – 14 = 26c, for some int c. 4a – 7 = 26c, for some int c 2a – 7 = 13c, so 7 = 4a – 26c 2a ≡ 7 (mod 13) is all we can 7 = 2(2a – 13c), which is impossible ascertain, the following above since 7 is NOT divisible by 2. implies that a ≡ 10 (mod 13), which can be determined by multiplying through by 7.
