Recall that an integer m is a square if m= k² for some integer k and is a triangular number if m = n(n+1)/2 for some integer n. Determine how many integers are simultaneously square and triangular.

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: Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing…

A: Since you have asked multiple questions, we will solve the 1st question for you. If you want any…

Q: Let B = [-1 1 4 -3] Find -4B

A: Solution

Q: Suppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible…

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Q: antip . Prove that 4" -3"-7 is a multiple of 84 for each positive even m.

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

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

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Q: Simplity. Assume all exponents are positive integers. 35n 45x² 2. 35n² 25x 1. 4. p²-3p-54 p-9 5.…

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Q: 2 4. Express 989 = 23 43 in the form x² + xy + 5y².

A: The given number is 989. The objective is to express the given number in the form of x2+xy+5y2.

Q: Let Find formulas for the entries of M", where n is a positive integer. Hint: a formula such as 5…

A: Given that the matrixM=16-147-5

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: 2. Prove the statement: The sum of the squares of any two consecutive integers is odd.

A: I have assumed that n be any integer. Then i have found the sum of square of two consecutive…

Q: 3) Find integers x and y such that 13x + 15y = 1.

A: We see, 15 - 13 = 2              => k(15 - 13) = 2k for any integer k              => 15 k -…

Q: Let Find formulas for the entries of M", where n is a positive integer. Hint: a formula such as 5…

A: The given problem is to find the entries of Mn for the given following matrix M. We have to find the…

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: We say two numbers m and n are consecutive perfect squares when m is the square of a positive…

A: We need to find year when men was born.

Q: Show that if 9 | a3 + b3 + c3 then 3 divides at least one of the integers a, b, or c.

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Q: For each positive integer n, define an = ((2+√3)" + (2-√3)") 2 It is known that an is a positive…

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Q: a. compute: 3. 5 2k b. compute: II k² +1 k=1

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