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13 Σεπ 2015 · Practice Problem Solutions 1. Given that 5x 6 (mod 8), nd x. [Solution: 6] 2. Find the last digit of 7100 [Solution: 1] 7100 (72) 50 49 ( 1)50 1 mod 10. 3. (1992 AHSME 17) The two-digit integers form 19 to 92 are written consecutively to form the large integer N = 192021 909192. Suppose that 3k is the highest power of 3 that is a factor of N ...
Ex 6: We can solve the equation 3 · x + 6 ≡ 8(mod 10) by using the sum (3) and multiplication (4) rules along with the above table: 3·x+6 ≡ 8(mod 10) =⇒
24 Μαΐ 2024 · Modular arithmetic, also known as clock arithmetic, deals with finding the remainder when one number is divided by another number. It involves taking the modulus (in short, ‘mod’) of the number used for division. If ‘A’ and ‘B’ are two integers such that ‘A’ is divided by ‘B,’ then:
This modulo calculator is a handy tool if you need to find the result of modulo operations. All you have to do is input the initial number x and integer y to find the modulo number r, according to x mod y = r. Read on to discover what modulo operations and modulo congruence are, how to calculate modulo and how to use this calculator correctly.
Basic Practice. Compute the modular arithmetic quantities, modulo n, in such a way that your answer is an integer 0 k < n. Do NOT use a calculator. Do these in your head. Compare to the answer key at the end.
Proof If x3+ 10000 = y3then x3+ 10000 ≡ y3(mod 7) (by Theorem 2.1.3(1)). Since 10000 ≡ 4 (mod 7), x3+4 ≡ y3(mod 7). But x3≡ −1,0, or 1 (mod 7) by previous example, so x3+ 4 ≡ 3,4 or 5 (mod 7), while y3≡ −1,0, or 1 (mod 7) contradiction. This example illustrates one of the uses of modular arithmetic.
Problem 1: just an example of previous identity. Find the remainder of 46 23 (a.k.a. 46 .23) on division by 7. Solution. 46 4 (mod 7) and 23 2 (mod 7) so 46 .23 4 .2 8 1 (mod 7). Problem 2: look for a small partner. Compute 138 (mod 7). What does this mean? The unique remainder r satisfying 0 r < 7. Easiest solution. 138 ( 1)8 1 (mod 7).
