(a) Let n be a positive integer. Prove that n is congruent to the sum of its digits modulo 9. Hint: first consider an example such as 345 = 3· 102 + 4· 10 +5... (b) Is the integer 123456789 divisible by 9?

Q: Find an integer between 0 and 24 that each of the following is equivalent to mod 25 (a) 500021…

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Q: 2n-1 b) Prove that if , r #1 (mod 2"), then n is even.

A: Given that x2n-1≢1mod2n . The objective is to show that n is even. The x2n-1≢1mod2n can be written…

Q: Let x be an integer coprime to 1450 that is not congruent to 1 modulo 1450. Suppose we know that x1…

A: The given two MCQs are exactly same. Hence we will just solve the first one. And we will use the…

Q: Evaluate (3459874 * 457832 * 34568) mod 10. Box in your final answer. Remember to show all work.

A: As we know  a*b mod n = (a mod n)*(b mod n)

Q: (a) 55 mod 7. (b) –101 div 3.

A: (a) 55 mod 7  As 55 = 7•7+6 So that  55 mod 7 = 6 mod 7

Q: Give a definition of an inverse of a number a modulo m.

A: if the two integer a and m modulo then  let x is an integer value so the x is inverse of an only if…

Q: Name all integers from -100 to 100 that are congruent to -1 modulo 25!

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Q: 3. Which of the following integers are divisible by 3? Of those that are, which are divisible by 9?…

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Q: Prove: If the sum of the digits of a number in base bis divisible by b 1, so is the number.

A: Given statement: Sum of digits of any number (x) in any base b is divisible by b-1. To prove: The…

Q: Match each of the following integers with its equivalent representation in a different base. (123) g…

A: Number system are of the type  Binary decimal with two bits 0 or 1. The decimal number is from 0 to…

Q: (c) Find 83-1 mod 105. (Note: Answer must be in between 0 and 104, inclusive.)

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Q: 5. Find the highest power of 2 that divides each of the following integers. a) (101111110)2 b)…

A: The decimal number is equal to the sum of binary digits (dn) times their power of 2 (2n): decimal =…

Q: (242²-121²) 242*121 O 1 O 2/3 O 11/3 O 3/2 ||

A: This question is based on formula  a2 - b2 = (a-b)*(a+b)

Q: Let n (arak-1 ·*a1a0)3 be the base 3 expansion of a positive integer n. Find the base 3 expansion of…

A: Solution: We know that if for any positive integer r have the base 3 expansion (ap,ap-1,.....,a0)3…

Q: Secton 5.1 Excerciser # 4

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Q: Discrete Math: Find the largest negative value of x such that x = 2 mod 3, x = 3 mod 5, x = 2 mod 7.

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Q: Najabhai

A: The objective of the question is to find all integers n that satisfy the equation n² + 503 = 0 (mod…

Q: 5. Find the highest power of 2 that divides each of the following integers. a) (101111110)2 b)…

A: Given: (101111110)2 (1010000011)2 (111000000)2 (1011011101)2 Which of these is divisible by 3

Q: If n is an integer, then (2n² + 3n)# 3(mod5).

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Q: The binary expansion 0.abcd... means 23 24 and each binary digit a, b, c, d, ... is either 0 or 1.…

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Q: Prove that 9? −1 is divisible by 8,for all n ≥ 1

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Q: The binary integer 1111 0110 1101 00012 is F7D116 in hexadecimal notation. Select one: O True O…

A: The question is solved with explanation in step 2.

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.