22. Suppose c is an integer. If c mod 15 = 3, what is10c mod 15? In other words, if division of c by 15 gives aremainder of 3, what is the remainder when 10c is divided%3Dby 15?

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Answered: 22. Suppose c is an integer. If c mod…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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If x mod 7 = 2, then x302 mod 7 D 4 D 5 2.
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solve the question 5 with explanation asap within 10 mins or else downvote
1. Consider a 4 digit number abcd and show that abcd−(a+b+c+d) is alwaysdivisible by 9. (Ex 1234 − 10 = 1224 = 9 · 136). 2. (A) How many different three-letter initials can a person have?(B) What if the initials are not repeated?(C) What if only consecutive initials are not repeated? ( please just answer part c on this one.. A and B are just so you know the entire question)
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Suppose c is any integer. If c mod 15 = 3, what is 10c mod 15? In other words, if division of c by 15 gives a remainder of 3, what is the remainder when 10c is divided by 15? Your solution should show that you obtain the same answer no matter what integer you start with.
5. Look at some decimal expressions of fractions such as 325' 5' 15' 20' 14' 50 1 7 9 16 71 50 91 by carrying out long divisions or using your calculator (Careful, calculators can give mis- leading results!), and see which numbers have a terminating decimal expression, while paying attention to the prime factorisation of the denominator. Hint: Take a terminating decimal expression and see what happens when you convert it to a fraction in lowest terms. (Don't include this part in your homework submission.) Now come up with a conjecture for which fractions have a terminating decimal expres- sion. Your conjecture should be of the following form: Let a, b EN with b > 1 and with gcd(a, b) = 1. Then a has a terminating decimal expression iff .... Now prove your conjecture. Note that this is an “if and only if” statement, requiring proofs for both implications.
Problem 4. Let n be any positive integer. Prove that there exists a positive integer k (depending on n) such that the following list of n consecutive integers: k, k + 1, ·, k + n - 1 contains no prime number at all. = Hint. Use the factorial (but k n! is NOT the correct answer, start from this and try to see what are missing). You also need the 2-out-of-3 property of division.
Informal Exercise 41. Make a multiplication table for Zg. Use it to find Zž. List all the inverses of all the units. To save time, you do not have to use bars or brackets in the tables. Hint: remember how to find remainders modulo 9, and use the fact that multiplication is commutative.
5. Let a and b be positive integers such that gcd(a, b+1) = 1 and b2 = 1 (mod a).What is the remainder in the division of b by a? Justify your answer.
3. What is the coefficient of ?9?11 in the expansion (?+?)20.

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22. Suppose c is an integer. If c mod 15 = 3, what is 10c mod 15? In other words, if division of c by 15 gives a remainder of 3, what is the remainder when 10c is divided %3D by 15?