6) If d +1 is the smallest positive divisor of an integer n, then d is prime. a) True b) False O a) True O b) False

Q: Use induction to prove that the sum of all the odd integers 1, 3, 5, 7, n to any odd integer n is a…

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Q: Find the number of positive integer divisors of a) 36 b) 99 c) 144 d) 2-3-5-7-11-13-17-19 e)…

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Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4т + 3 бт + 1

A: Step:-1 Given integer form is 12 k +11 Take  a =12 k +11  then a=12 k +11⇒a=12 k +8+3⇒a=43k +2…

Q: 10. If j and k are both prime numbers, and j> k, which of the following statements must be true? I.…

A: Algebra problem....

Q: Show that if n is an integer and 3n³ + n is odd then n is odd, by using a) Proof by Contrapositive…

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Q: 7) Prove by contradiction: For all prime numbers a, b, and c, a² + b² + c².

A: To prove for all prime numbers a, b, and c, a² + b² not equal to c^2.

Q: a) Find integers a and b such that a + b =a-b (mod 5).

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Q: Prove or disprove: For every positive integer n, 22n – 1 is divisible by 3. -

A: I have proved the result using mathematical induction

Q: Any integer of the form 12k + 11 is also of the form а) Зт + 1 b) 5m + 2 4m + 3 бт + 1

A: We have to do proper manipulation.

Q: 2. Find the integer a such that a = 2*(mod31) and -16 < a< 15. а) 16 b) -16 с) -15 d) 15 e) 8

A: Here we have to find a.

Q: Write each as a single factorial. i) 90*8! ii) (n² + n)(n 1)! iii) (n +1)!/(n -1)!

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Q: Prove that 3 devides 17k+1-2*5k+3+11k+3whenever k is a non-negative integer.

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Q: 6) Let n be a positive integer such that 7 | (52n-1 +22n-1). Prove that 7 || (52n+1+22n+¹). (Hint:…

A: Since, given that  52n-1+ 22n-1  is divisible by 7 .  We have to prove that  52n+1+22n+1 is also…

Q: 7) Use a direct proof to show that: a) The sum of an even integer and an odd integer is odd. b) If n…

A: Use the following concepts, to obtain the required result. An even integer m can be written as a…

Q: 2.- Proof by induction that 4" - 1 is divisible by 3

A: Mathematical induction is a powerful proof technique used in mathematics to prove that a given…

Q: How many positive integers between 50 and 100 a) are divisible by 7? Which integers are these? b)…

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Q: 23 - where a and b are positive integers; How many b values are there for the condition 2a + 3b =…

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Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

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Q: Prove by contraposition: if a, b and e are integers and be is not divisible by a, then b is not…

A: Given a,b and c are integers. Statement : P: bc is not divisible by a. Statement: Q: b is not…

Q: Show that if n is an integer and n³ + 5 is odd, then n even using a) a proof by contraposition. b) a…

A: Proofs: The proof is a valid argument that establishes the truth of a statement. Types of Proofs:…

Q: 3. Use induction to prove that 15"+2 + 162n+1 is divisible by 241 for every nonnegative integer п.

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Q: 1) Provide a counter example to the following statements: a) If a|b then b|a. b) If n is prime then…

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Q: 5. Use the direct proof to show that " If n is an odd integer, then n2 + 3 is even

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Q: Let i be a positive integer and A, = {[.., i – 6, i – 4, i – 2, i, i + 2,i + 4,1+ 6, . ). Then N 4…

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Q: prove or disprove If an integer n has no prime divisors smaller than itself, then it must be prime.

A: A prime number is an integer whose only divisors are 1 and the number itself.  Now, let n be a…

Q: only d)

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Q: ,,,,,,,

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Q: 7. Without using any theorems from the book, prove the following statement: Let n be an integer. If…

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Q: Use proof by contraposition to prove this.   If n is an integer and 3n is not a multiple of six,…

A: We prove that by contraposition If n is an integer and 3n is not a multiple of six, then n is odd.

Q: Which of the following is the sum of all four-digit numbers that can be formed by the digits 1, 3,…

A: The given digits are 1,3,5,7,9 We know sum of r digits number = n-1Pr-1 Sum of all n digits number…

Q: Prove the statement "If a and b are integers one number apart, then the sum a + b is even“ using:…

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