Tag: maths

Questions Related to maths

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

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

  1. $2p$ or $(4p-1)$

  2. $4p$ or $(4p-1)$

  3. $3p$ or $(3p+1)$

  4. $4p$ or $(4p+1)$

Show answer
Correct Option: D
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 $4p+1$

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?

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

  1. 2450

  2. 2550

  3. 2460

  4. 22500

Show answer
Correct Option: B
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 $=$ $2500+50=2550$

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

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

  1. $-9$

  2. $10$

  3. $-10$

  4. $-1$

Show answer
Correct Option: D
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 $-1$.

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

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

  1. $0$

  2. $2$

  3. $-2$

  4. $3$

Show answer
Correct Option: A
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 $0$.

The 6th consecutive odd integer after $-5$ is

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

  1. $0$

  2. $-10$

  3. $-13$

  4. $7$

Show answer
Correct Option: D
Explanation:

$-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9$


$6$th consecutive odd integer after $-5$, as seen from above is $7$

Addition of largest odd number and smallest even number from the integers $-5$ to $5$ is

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

  1. $9$

  2. $-9$

  3. $1$

  4. $-1$

Show answer
Correct Option: C
Explanation:

$-5,-4,-3,-2,-1,0,1,2,3,4,5$


Largest odd number $= 5$
Smallest even number $= -4$

$\therefore$ Answer $= 5 + (-4) = 1$

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

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

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

  2. $-13,-11,-9$

  3. $-15,-13,-11$

  4. $-11,-9,-7$

Show answer
Correct Option: D
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 $-11,-9,-7$.

Sum of one odd and one even integers is :

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

  1. Even

  2. Odd

  3. Both

  4. Can't be determined

Show answer
Correct Option: B
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.

Sum of two even integers is :

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

  1. Even

  2. Odd

  3. Both

  4. Can't be determined

Show answer
Correct Option: A
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.

If n is an integer, which of the following cannot be odd?

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

  1. $n+3$

  2. $n+1$

  3. $2n$

  4. $3n$

Show answer
Correct Option: C
Explanation:

We know $2n=2\times n$

So, $2n$ is a multiple of $2$. Hence, it can not be an odd number.
It is always even.