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Modular Arithmetic Practice - CMU

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Modular Arithmetic PracticeJoseph ZollerSeptember 13, 2015Practice Problem Solutions1. Given that 5x 6 (mod 8), findx.[Solution: 6]2. Find the last digit of 7100[Solution: 1]7100 (72)50 4950 ( 1)50 1 mod (1992 AHSME 17) The two-digit integers form 19 to 92 are written consecutively to form thelarge integerN= 192021 that 3kis the highest power of 3 that is a factor ofN. What isk?[Solution:k= 1]We know thatN S(N) mod (N) = 1+9+9+0+9+1+9+2+10(2+...+8)+7(0+...9) =40 + 10(35) + 7(45) = 40 + 350 + 315 = 705. ThenN S(N) S(S(N)) S(705) 12 3mod 9. Thus, it is only divisible by 3 and not 9, andk= (2000 AMC 12 18) In yearN, the 300th day of the year is a Tuesday. In yearN+ 1, the 200thday is also a Tuesday. On what day of the week did the 100th day of the yearN 1 occur?[Solution: Thursday]There are either 65 + 200 = 265 or 66 + 200 = 266 days between the first two dates dependingupon whether or not yearNis a leap year. Since 7 divides into 266, then it is possible forboth dates to Tuesday; hence yearN+ 1 is a leap year andN 1 is not a leap year.

Sep 13, 2015 · Modular Arithmetic Practice Joseph Zoller September 13, 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.

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