A matrix M has 3 rows and 4 columns. The 12 entries in the

For each of the values of n and d given in 1–6, find integers q and r such that n = dq + r and 0 \(\leq\) r < d. n = 62, d = 7 Text Transcription: leq

Problem 53E Prove that if m, d, and k are integers and d > 0, then (m + dk) mod d = m mod d.

For each of the values of n and d given in 1–6, find integers q and r such that n = dq + r and 0 \(\leq\) r < d. n = 70, d = 9 Text Transcription: leq

For each of the values of n and d given in 1–6, find integers q and r such that n = dq + r and 0 \(\leq\) r < d. n = 36, d = 40 Text Transcription: leq

For each of the values of n and d given in 1–6, find integers q and r such that n = dq + r and 0 \(\leq\) r < d. n = -45, d = 11 Text Transcription: leq

For each of the values of n and d given in 1–6, find integers q and r such that n = dq + r and 0 \(\leq\) r < d. n = 3, d = 11 Text Transcription: leq

Evaluate the expressions in 7–10. a. 43 div9 b. 43 mod 9

For each of the values of n and d given in 1–6, find integers q and r such that n = dq + r and 0 \(\leq\) r < d. n = -27, d = 8 Text Transcription: leq

Evaluate the expressions in 7–10. a. 50 div 7 b. 50 mod 7

Evaluate the expressions in 7–10. a. 28 div 5 b. 28 mod 5

Evaluate the expressions in 7–10. a. 30 div 2 b. 30 mod 2

Justify formula (4.4.1) for general values of DayT and N.

Problem 14E If today is Tuesday, what day of the week will it be 1,000 days from today?

Problem 15E January 1, 2000, was a Saturday, and 2000 was a leap year. What day of the week will January 1, 2050, be?

Problem 13E On a Monday a friend says he will meet you again in 30 days. What day of the week will that be?

Problem 16E Suppose d is a positive integer and n is any integer. If d | n, what is the remainder obtained when the quotient-remainder theorem is applied to n with divisor d?

Check the correctness of formula (4.4.1) given in Example 4.4.3 for the following values of DayT and N. a. DayT = 6 (Saturday) and N = 15 b. DayT = 0 (Sunday) and N = 7 c. DayT = 4 (Thursday) and N = 12

Problem 17E Prove that the product of any two consecutive integers is even.

Problem 18E The result of exercise suggests that the second apparent blind alley in the discussion of Example 1 might not be a blind alley after all. Write a new proof of Theorem based on this observation. Exercise Prove that the product of any two consecutive integers is even. Example 1 The Square of an Odd Integer Prove: The square of any odd integer has the form 8m + 1 for some integer m. Solution Begin by asking yourself, “Where am I starting from?” and “What do I need to show?” To help answer these questions, introduce variables to represent the quantities in the statement to be proved. Formal Restatement:? odd integers n, ? an integer m such that n2 = 8m + 1. From this, you can immediately identify the starting point and what is to be shown. Starting Point:Suppose n is a particular but arbitrarily chosen odd integer. To Show:? an integer m such that n2 = 8m + 1. This looks tough. Why should there be an integer m with the property that n2 = 8m + 1? That would say that (n2 ? 1)/8 is an integer, or that 8 divides n2 ? 1. Perhaps you could make use of the fact that n2 ? 1 = (n ? 1)(n + 1). Does 8 divide (n ? 1)(n + 1)? Since n is odd, both (n ? 1)and (n + 1)are even. That means that their product is divisible by 4. But that’s not enough. You need to show that the product is divisible by 8. This seems to be a blind alley. You could try another tack. Since n is odd, you could represent n as 2q + 1 for some integer q. Then n2 = (2q + 1)2 = 4q2 + 4q + 1 = 4(q2 + q)+ 1. It is clear from this analysis that n2 can be written in the form 4m + 1, but it may not be clear that it can be written as 8m + 1. This also seems to be a blind alley.* Yet another possibility is to use the result of Example 2. That example showed that any integer can be written in one of the four forms 4q, 4q + 1, 4q + 2, or 4q + 3. Two of these, 4q + 1 and 4q + 3, are odd. Thus any odd integer can be written in the form 4q + 1 or 4q + 3 for some integer q. You could try breaking into cases based on these two different forms It turns out that this last possibility works! In each of the two cases, the conclusion follows readily by direct calculation. The details are shown in the following formal proof: Theorem The square of any odd integer has the form 8m + 1 for some integer m. Proof: Suppose n is a [particular but arbitrarily chosen] odd integer. By the quotient-remainder theorem, n can be written in one of the forms 4q or 4q +1 or 4q +2 or 4q + 3 for some integer q. In fact, since n is odd and 4q and 4q + 2 are even, n must have one of the forms 4q +1 or 4q + 3. Case 1 (n=4q+1 for some integer q):[We must find an integer m such that n2 = 8m + 1.] Since n = 4q + 1, Let m = 2q2 + q. Then m is an integer since 2 and q are integers and sums and products of integers are integers. Thus, substituting, n2 = 8m + 1 where m is an integer. Case 2 (n=4q +3 for some integer q):[We must find an integer m such that n2 = 8m + 1.] Since n = 4q + 3, [The motivation for the choice of algebra steps was the desire to write the expression in the form 8•(some integer) + 1.] Let m = 2q2 + 3q + 1. Then m is an integer since 1, 2, 3, and q are integers and sums and products of integers are integers. Thus, substituting, n2 = 8m + 1 where m is an integer. Cases 1 and 2 show that given any odd integer, whether of the form 4q + 1 or 4q + 3, n2 = 8m + 1 for some integer m. [This is what we needed to show.] Example 2 Representations of Integers Modulo 4 Show that any integer can be written in one of the four forms n = 4q or n = 4q +1 or n = 4q +2 or n = 4q + 3 for some integer q. Solution Given any integer n, apply the quotient-remainder theorem to n with d = 4. This implies that there exist an integer quotient q and a remainder r such that n = 4q + r and 0 ? r<4. But the only nonnegative remainders r that are less than 4 are 0, 1, 2, and 3. Hence n = 4q or n = 4q +1 or n = 4q +2 or n = 4q + 3 for some integer q.

Prove that for all integers \(n, n^{2}-n+3\) is odd. Text Transcription: n, n^2-n+3

Problem 21E Suppose b is an integer. If b mod 12 = 5, what is 8b mod 12? In other words, if division of b by 12 gives a remainder of 5, what is the remainder when 8b is divided by 12?

Prove that for all integers n, if n mod 5 = 3 then \(n^{2}\) mod 5 = 4. Text Transcription: n^2

Problem 22E 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 by 15?

Problem 20E Suppose a is an integer. If a mod 7 = 4, what is 5a mod 7? In other words, if division of a by 7 gives a remainder of 4, what is the remainder when 5a is divided by 7?

Problem 24E Prove that for all integers m and n, if m mod 5 = 2 and n mod 5 = 1 then mn mod 5 = 2.

Problem 25E Prove that for all integers a and b, if a mod 7 = 5 and b mod 7 = 6 then ab mod 7 = 2.

Problem 26E Prove that a necessary and sufficient condition for a nonnegative integer n to be divisible by a positive integer d is that n mod d = 0.

Show that any integer n can be written in one of the three forms n = 3q or n = 3q +1 or n = 3q + 2 for some integer q.

a. Use the quotient-remainder theorem with d = 3 to prove that the square of any integer has the form 3k or 3k + 1 for some integer k. b. Use the mod notation to rewrite the result of part (a).

a. Use the quotient-remainder theorem with d = 3 to prove that the product of any three consecutive integers is divisible by 3. b. Use the mod notation to rewrite the result of part (a).

In 31–33, you may use the properties listed in Example 4.2.3. a. Prove that for all integers m and n,m + n and m ? n are either both odd or both even. b. Find all solutions to the equation \(m^{2}-n^{2}\) = 56 for which both m and n are positive integers. c. Find all solutions to the equation \(m^{2}-n^{2}\) = 88 for which both m and n are positive integers. Text Transcription: m^2-n^2

In 31–33, you may use the properties listed in Example 4.2.3. Given any integers a, b, and c, if a ? b is even and b ? c is even, what can you say about the parity of 2a ? (b + c)? Prove your answer.

a. Use the quotient-remainder theorem with d = 3 to prove that the product of any two consecutive integers has the form 3k or 3k + 2 for some integer k. b. Use the mod notation to rewrite the result of part (a).

In 31–33, you may use the properties listed in Example 4.2.3. Given any integers a, b, and c, if a ? b is odd and b ? c is even, what can you say about the parity of a ? c? Prove your answer.

Problem 34E Given any integer n, if n > 3, could n, n + 2, and n + 4 all be prime? Prove or give a counterexample.

Prove each of the statements in 35–46. The product of any four consecutive integers is divisible by 8.

Prove each of the statements in 35–46. The fourth power of any integer has the form 8m or 8m + 1 for some integer m.

Prove each of the statements in 35–46. The square of any integer has the form 4k or 4k + 1 for some integer k.

Prove each of the statements in 35–46. The sum of any four consecutive integers has the form 4k + 2 for some integer k.

Prove each of the statements in 35–46. For any integer n, n\(\left(n^{2}-1\right)\)(n + 2) is divisible by 4. Text Transcription: (n^2-1)

Prove each of the statements in 35–46. For all integers m, \(m^{2}\) = 5k, or \(m^{2}\) = 5k + 1, or \(m^{2}\) = 5k + 4 for some integer k. Text Transcription: m^2

Prove each of the statements in 35–46. Every prime number except 2 and 3 has the form 6q + 1 or 6q + 5 for some integer q.

Prove each of the statements in 35–46. If n is an odd integer, then \(n^{4}\) mod 16 = 1. Text Transcription: n^4

Prove each of the statements in 35–46. For all real numbers r and c with c \(\geq\) 0, if ? c \(\leq\) r \(\leq\) c, then |r| \(\leq\) c. Text Transcription: geq leq

Prove each of the statements in 35–46. For all real numbers x and y, \(|x| \cdot|y| = |x y|\). Text Transcription: |x| cdot |y| = |x y|

Prove each of the statements in 35–46. For all real numbers r and c with c \(\geq\) 0, if |r| \(\leq\) c, then ? c \(\leq\) r \(\leq\) c. Text Transcription: geq leq

Prove each of the statements in 35–46. For any integer n, \(n^{2}\) + 5 is not divisible by 4. Text Transcription: n^2

Let M be a matrix with m rows and n columns, and suppose that the entries of M are stored in a computer’s memory in row major form (see exercise 47) in locations N, N + 1, \(N+2, \ldots, N+m n-1\). Find formulas in k for r and s so that \(a_{rs}\) is stored in location N + k. Text Transcription: N+2, ldots, N+m n-1 a_rs

A matrix M has 3 rows and 4 columns. \(\left[\begin{array}{llll} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \end{array}\right]\) The 12 entries in the matrix are to be stored in row major form in locations 7,609 to 7,620 in a computer’s memory. This means that the entries in the first row (reading left to right) are stored first, then the entries in the second row, and finally the entries in the third row. a. Which location will \(a_{22}\) be stored in? b. Write a formula (in i and j ) that gives the integer n so that \(a_{ij}\)j is stored in location 7,609 + n. c. Find formulas (in n) for r and s so that \(a_{r s}\) is stored in location 7,609 + n. Text Transcription: [a_11 a_12 a_13 a_14 a_21 a_22 a_23 a_24 a_31 a_32 a_33 a_34] a_22 a_ij a_rs

Problem 49E If m, n, and d are integers, d > 0, and m mod d = n mod d, does it necessarily follow that m = n? That m ? n is divisible by d? Prove your answers.

Problem 50E If m, n, and d are integers, d > 0, and d | (m ? n), what is the relation betweenm mod d and n mod d? Prove your answer.

Problem 51E If m, n, a, b, and d are integers, d > 0, and m mod d = a and n mod d = b, is (m + n) mod d = a + b? Is (m + n) mod d = (a + b) mod d? Prove your answers.

Problem 52E If m, n, a, b, and d are integers, d > 0, and m mod d = a andn mod d = b, is (mn) mod d =ab? Is (mn) mod d =abmod d? Prove your answers.