Use a direct proof to show that every odd integer is a difference of two squares. (For example: 1 = 12 – 02 and 5 = 32 – 22, and so on.)

Q: aj = 9 and an = - — Зап-1 5 then find the value of a5. -

A: Given query is to find the value of a5 .

Q: The gcd of 687 and 24 is: 01 O 11 12 3

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Q: For each of the following numbers N, compute the values of N + 12, N + 22, N + 32, N + 42, … until…

A: Factor  into prime factors.

Q: Show that if a positive integer is odd, then it's square is odd.

A: To show that if a positive integer is odd then its square is odd

Q: Show that if n is any integer, then 3 divides n3 − n

A: To prove:  if n is any integer, then 3 divides n3 − n. Concept used: Here we shall use the identity:…

Q: you’re thinking of a number 1-50 and i have to guess what it is. i asked these questions in this…

A: Given: A numbers between 1-50. To find: Find a number greater than 25, divisible by 2, 3 and 5 and…

Q: at any positive odd integer is of the

A: Here we have to show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5,…

Q: Prove that for a positive integer n we have that 33n+3 – 26n – 27 is divisible by 169. - -

A: The given expression is: 33n+3-26n-27 This is to be proved that the expression is divisible by 169.

Q: Given any real numbers a and d, consider the expression (a + md) + (a + (m + 1)d) + (a + (m + 2)d) +…

A:

Q: Prove the following: There exist no integers a and b for which 14a + 4b = 1. Please explain the…

A:

Q: (b) If a, b and c are integers such that b| a and b| c, then b | (2a – 3c).

A: We have to proof

Q: Give the primep such that 4lp +1 is a complete square:

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Q: o prove that 99 is not the square of a positive integer, we just need to consider three cases for an…

A: The objective is to determine if the given statement is true or false. The statement is  to prove…

Q: 1. Prove that the sum of an integer and its square is even

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Q: (d) (11 × 14" + 1) is a composite integer for every positive integer

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Q: Prove that the sum of the squares of two odd integers cannot be the square of an integer.

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Q: "Every positive integer can be written as the sum of the square of 3 positive integers." prove…

A: Given: "Every positive integer can be written as the sum of the square of 3 positive integers." To…

Q: let a = 3^4 7^3 11^2 13^5 and b = 2^6 5^2 7^2 11^3 13^4 Determine the values of (a,b) and [a,b] and…

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Q: Find the formula for 1+2-3+4-5...

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Q: Give a direct proof of the following statement. If x is an even integer, then x² – 6x + 5 is odd. |

A: Remark: We know that addition or subtraction of an even and an odd integer is always an odd integer.…

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Which of the following quadratics can be factored using the "difference of squares"? 36x²-81 27-3x²…

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Q: For positive integers a and b, show that a ab

A: Consider the provided question, For positive integers a and b,  we need to show that, xab=xab

Q: (d) Prôvé that the square of every prime number greater than 3 is one more than a multiple 24.

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Q: (b) Consider the following set of real numbers 2n 4-{²₂²REN} :ne n+2 A = Answer the following…

A: We have to find the supremum of

Q: Show that V3 +2/6 is an irrational number.

A: A rational number can be represented as pq, where p, q are integers and q≠0. If a number can not be…

Q: #5 Find An if n is a positive integer and A = 4 -3 - 2

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Q: square of any odd integer is od.

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Q: Which of the following trinomials can be factored into the perfect square (7x- 10)2? O 49x2 - 140x -…

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Statements and Logic

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