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Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

If $p$ is an integer, then every square integer is of the form

  1. $2p$ or $(4p-1)$
  2. $4p$ or $(4p-1)$
  3. $3p$ or $(3p+1)$
  4. $4p$ or $(4p+1)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Let $x$ be any positive integer.
For example Let $x=2\Rightarrow\,{x}^{2}=4$ is a square number is of the form $4p$ where $p\in Z$
Let $x=3\Rightarrow\,{x}^{2}=9=2\times 4+1$ is a square number is of the form $4p+1$ where $p\in Z$
Let $x=4\Rightarrow\,{x}^{2}=16$ is a square number is of the form $4p$ where $p\in Z$
Let $x=5\Rightarrow\,{x}^{2}=25=4\times 6+1$ is a square number is of the form $4p+1$ where $p\in Z$
Let $x=4p\Rightarrow\,{x}^{2}=16{p}^{2}=4\left(4{p}^{2}\right)=4q$ where $q=4{p}^{2}$ is a square number is of the form $4p$ where $p\in Z$
If $x=4p+1\Rightarrow\,{x}^{2}={\left(4p+1\right)}^{2}=16{p}^{2}+8p+1=4\left(4{p}^{2}+2p\right)+1=4q+1$ where $q=4{p}^{2}+2p$
If $x=4p+3\Rightarrow\,{x}^{2}={\left(4p+3\right)}^{2}=16{p}^{2}+24p+9=4\left(4{p}^{2}+6p+2\right)+1=4q+1$ where $q=4{p}^{2}+6p+2$
$\therefore\,$in each of the above cases,${x}^{2}$ is of the form $4p$ or $4p+1$
Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

Given that the sum of the odd integers from $1$ to $99$ inclusive is $2500$, what is the sum of the even integers from $2$ to $100$ inclusive?

  1. 2450

  2. 2550

  3. 2460

  4. 22500
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Odd integers $=1,3,5,7,9.....$

Sum of odd integers $=2500$
Even integers $=2,4,6,8,,10......$
In the series of even integers each term is one more than the each term of odd integers.
Hence, there are $50$ terms.
So, the sum of even integers $=$ $2500+50=2550$

Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

The largest odd integer from  $-10$ to $0$ is:

  1. $-9$
  2. $10$
  3. $-10$
  4. $-1$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
An odd number is an integer which is not divisible by two. If it is divided by two, then the result is a fraction. The set of odd integers is $-11,-9,-7,-5,-3,-1,1,3,5,7,.....$

The following is the set of odd integers between $-10$ and $0$:

$-9,-7,-5,-3,-1$ where $-9<-7<-5<-3<-1$

Therefore, $-1$ is the largest.

Hence, the largest odd integer from $-10$ and $0$ is $-1$.
Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

Addition of odd integers between  $-3 \ and\ 3$ is

  1. $0$
  2. $2$
  3. $-2$
  4. $3$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
An odd number is an integer which is not divisible by two. If it is divided by two, then the result is a fraction. The set of odd integers is $-5,-3,-1,1,3,5,7,.....$

The following is the set of odd integers between $-3$ and $3$:

$-1,1$

Therefore, the sum is $-1+1=0$

Hence, addition of odd integers between $-3$ and $3$ is $0$.
Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

Find three consecutive odd integers such that the sum of first and third integers is same as the second integer when decreased by $9$.

  1. $-9,-7,-5$
  2. $-13,-11,-9$
  3. $-15,-13,-11$
  4. $-11,-9,-7$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Let us say the first odd integer be $x$. The second consecutive odd integer would be $x+2$ (zit would not be $x+1$ because that would result in an even integer. The sum of two odd integers is even). The third consecutive odd integer would be $(x+2)+2$ or $x+4$.


Now, it is given that the sum of first and third integers is same as the second integer when decreased by $9$ which means:

$x+(x+4)=(x+2)-9\ \Rightarrow 2x+4=x-7\ \Rightarrow 2x-x=-7-4\ \Rightarrow x=-11$

Therefore, the first odd integer is $-11$ then the second integer is $x+2=-11+2=-9$ and the third integer is $x+4=-11+4=-7$

Hence, the three consecutive odd integers are $-11,-9,-7$.

Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

Sum of one odd and one even integers is :

  1. Even

  2. Odd

  3. Both

  4. Can't be determined
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

An even integer is an integer that is evenly divisible by $2$, that is, division by $2$ results in an integer without any remainder. The set of even integers is:


$.....-8,-6,-4,-2,2,4,6,8,.....$

Whereas, an odd integer is an integer that is not divisible by $2$. The set of odd integers is:

$.....-9,-7,-5,-3,-1,1,3,5,7,9.....$

Now let us take an even integer say, $2$ and an odd integer say $3$, then their sum will be $2+3=5$ which is an odd integer.

Hence, the sum of one odd and one even integer is always odd.

Multiple choice maths negative numbers and integers even and odd numbers sum of numbers odd and even numbers

Sum of two even integers is :

  1. Even

  2. Odd

  3. Both

  4. Can't be determined
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

An even integer is an integer that is evenly divisible by $2$, that is, division by $2$ results in an integer without any remainder. The set of even integers is:


$.....-8,-6,-4,-2,2,4,6,8,.....$

Now let us take two even integers say, $2$ and $4$, then their sum will be $2+4=6$ which is also an even integer.

Hence, the sum of even integers is always even.