### Understanding Integer Classifications and Statement Negation#### 1. Integer ClassificationsAssume that $ t $ is a particular integer. Use the definitions of even, odd, or composite to justify the following statements:(a) **35 is an odd integer.** - Justification: An odd integer is not divisible by 2. Since $ 35 \div 2 = 17.5 $, which is not an integer, 35 is odd.(b) **-14 is an even integer.** - Justification: An even integer is divisible by 2. Since $ -14 \div 2 = -7 $, which is an integer, -14 is even.(c) **77 is a composite integer.** - Justification: A composite integer has more than two distinct positive divisors. The divisors of 77 are 1, 7, 11, and 77, thus making it composite.(d) **$ 14t - 8 $ is an even integer.** - Justification: $ 14t $ (where 14 is an even number and $ t $ is any integer) is even, and subtracting 8 (also even) from it yields another even integer.(e) **$ 8t + 9 $ is an odd integer.** - Justification: $ 8t $ (where 8 is even) is even for any integer $ t $, and adding 9 (which is odd) results in an odd integer.#### 2. Statement Negation and DisproofConsider the statement for all real numbers $ x $, $ x^2 > 0 $.(a) **Write the negation of the statement above.** - Negation: There exists a real number $ x $ such that $ x^2 \leq 0 $.(b) **Disprove the original statement by giving a counterexample.** - Counterexample: Consider $ x = 0 $. Here, $ 0^2 = 0 $, which is not greater than 0, thereby disproving the statement $ x^2 > 0 $ for all $ x $.

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Description: ### Understanding Integer Classifications and Statement Negation#### 1. Integer ClassificationsAssume that $ t $ is a particular integer. Use the definitions of even, odd, or composite to justify the following statements:(a) **35 is an odd integer

Even number are those numbers which is divisible by 2. If the unit digit of a number is 0,2,4,6 or 8…

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1. Assume that t is a particular integer. Use the definition of even, odd, or composite to justify the following. (a) 35 is an odd integer. (b) -14 is an even integer. (c) 77 is a composite integer.

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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### Understanding Integer Classifications and Statement Negation

#### 1. Integer Classifications

Assume that $ t $ is a particular integer. Use the definitions of even, odd, or composite to justify the following statements:

(a) **35 is an odd integer.**
- Justification: An odd integer is not divisible by 2. Since $ 35 \div 2 = 17.5 $, which is not an integer, 35 is odd.

(b) **-14 is an even integer.**
- Justification: An even integer is divisible by 2. Since $ -14 \div 2 = -7 $, which is an integer, -14 is even.

(c) **77 is a composite integer.**
- Justification: A composite integer has more than two distinct positive divisors. The divisors of 77 are 1, 7, 11, and 77, thus making it composite.

(d) **$ 14t - 8 $ is an even integer.**
- Justification: $ 14t $ (where 14 is an even number and $ t $ is any integer) is even, and subtracting 8 (also even) from it yields another even integer.

(e) **$ 8t + 9 $ is an odd integer.**
- Justification: $ 8t $ (where 8 is even) is even for any integer $ t $, and adding 9 (which is odd) results in an odd integer.

#### 2. Statement Negation and Disproof

Consider the statement for all real numbers $ x $, $ x^2 > 0 $.

(a) **Write the negation of the statement above.**
- Negation: There exists a real number $ x $ such that $ x^2 \leq 0 $.

(b) **Disprove the original statement by giving a counterexample.**
- Counterexample: Consider $ x = 0 $. Here, $ 0^2 = 0 $, which is not greater than 0, thereby disproving the statement $ x^2 > 0 $ for all $ x $.

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