MATHS M.C.Q. [1 Marks Each]

 The smallest prime number is $2$
So, Successor of $2 = 2 + 1 = 3$

$ 100$ million $= 100,000,000100,000,000$
$= 10,00,00,000 = 10 = 10,00,00,000 = 10$ crore

 We know that natural numbers start from $1, 2, 3, ...$
So, the smallest natural number $=1$

 Using distributive law of multiplication over subtraction, we get
$1735 \times 1232 - 1735 \times 232 = 1735 (1232 - 232)$
$= 1735 \times 1000$
$= 1735000$
Hence, the correct option is $(c).$

$ (-1) \times (-1) \times (-1).......$odd times $=$ Negative sign.

 A set of numbers which do not have any other common factor other than $1$ are called co-prime.
The $LCM$ of two co-prime numbers is equal to their product.
The $HCF$ of two co-prime numbers is $1.$

$\Rightarrow $ The smallest $4 −$ digit number $= 1000.$
$\Rightarrow $ The greatest $4 −$ digit number $= 9999.$
$\Rightarrow $ Total $4 −$ digit numbers $= (9999 - 1000) + 1$
$= 8999 + 1$
$= 9000$
So, we say that we have $9000$ four - digit numbers.

$67 + 33 = 33 + 67$ is an example of​ commutative property of addition.

Closure property in whole number states that if we do the operation between two whole number we should get a whole number. $2, 5$ are
whole number. $2 - 5 = (-3)$ is not a whole number. $\frac{2}{5} ​= 0.4$ is not a whole number.
Closure property is not satisfied w.r.t. subtraction and division in whole numbers.

$4 \times 538 \times 25$
$= 100 \times 538$
$= 53800$

The multiplicative identity property states that any time you multiply a number by $1,$ the result, or product, is that original number.
So it is clear that multiplicative property is given by

The additive identity element is $0.$ We know that the sum of $0$ and any number is the number itself.
Let the number be $x.$
$\therefore x + 0 = 0$
$\Rightarrow x = 0$

The sum of two odd numbers is an even number.
Example: $5 + 3 = 8$

Here, the given number is $28.$
Additive inverse of a given number is a number which when added to the given number gives zero.
Now $28 + (-28) = 0$
Thus, $-28$ is the additive inverse of $28.$
Hence, the correct option is $(c).$

 Whole numbers are not closed under subtraction operation because when any two whole numbers are considered and from them one is subtracted from the other, the difference so obtained is not necessarily a whole number.
Eg. $2 - 5 = -3.$

$\frac{4}{-17}+\frac{0}{-17} =\frac{4}{-17}$
$\frac{4}{-17}-\frac{0}{-17}= \frac{4}{-17}$
Therefore we use both $+$ or $-$

Natural numbers set is $1, 2, 3, ......$
Whole numbers set is $0, 1, 2, 3, .....$
Clearly every natural number is a whole number.

The natural numbers along with $0$ form the collection of whole numbers.
So, the numbers $0, 1, 2, 3, 4, …$ form the collection of whole numbers.
A natural number which is not divisible by $2$ is called an odd whole number.
So, in the collection $“0, 1, 2, 3, 4, …”, 1$ is the smallest odd whole number.
Hence, the correct option is $(b).$

The product of the multiplicative identity of whole number i.e.,
$1$ and a whole number we get the number itself.

$A = \{1, 2, 3, ........, n\} $
$B = \{a, b\}$
A has $n$ elements. $B$ has $2$ elements.
$\therefore$ No. of surjection is $ 2^{\text{n}}-2$

Whole numbers are not closed under subtraction operation because when any two whole numbers are considered and from them one is subtracted from the other,
the difference so obtained is not necessarily a whole number.
Eg. $2 - 5 = −3.$

 If we write natural numbers from $1$ to $100,$ the number of times the digit $5$ has been written is $20.$

The numbers $0, 1, 2, 3, 4, ….$ form the collection of whole numbers.
The smallest whole number is $0.$
So, $0$ does not have a predecessor.
Hence, the correct option is $(b).$

$x$ can have any values between $3$ and $4$ so it can be rational or irrational.
option $c = 0.9$ and options $A$ and $D$ are outside the given range.

Distributive property say that multiplication an division can be distributed over parenthesis.
$a \times (b + c) = a \times b + a \times c$

Natural numbers are the set of whole numbers $($except $0).$
The numbers $1,2,3,4,5.......$ are called natural numbers.

The Distributive Property is easy to remember if you recall that "multiplication distributes over addition".
Formally, they write this property as $"a(b + c) = ab + ac".$
$14 \times 6 + 14 \times 4$
$= 14 \times (6 + 4) ($distributive property$)$
$= 14 \times 10$
$= 140.$

$x(y + z) = (x \times y) + (x \times z)$ shows the distributive property of multiplication over addition.

Number of whole numbers $= (1203 - 1018) - 1$
$= 185 - 1$
$= 184$

$587 \times 99$
$= 587 \times (100 - 1)$
$= 587 \times 100 - 587 \times 1$
$= 58700 - 587$
$= 58113$

By using the distributive property, we have:
$24679 \times 92 + 24679 \times 8$
$= 24679 \times (92 + 8)$
$= 24679 \times 100$
$= 2467900$

We know that, the sum of $0$ and any rational number is the number itself.
Thus, $0$ is the additive identity for rational numbers.

$LCM$ of $15, 20, 24$ and $32$ is given by
$15 = 3 \times 5 = 3^1 \times 5^1$
$20 = 2 \times 2 \times 5 = 2^2 \times 5^1$
$24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3^1$
$32 = 2 \times 2 \times 2 \times 2 \times 2 = 2^5$
$LCM = 2^5 \times 3^1 \times 5^1 = 480$

Smallest whole number $= 0$
Greatest $2-$digit whole number $= 99$
The whole numbers between $0$ and $99$ are $1, 2, 3, 4 …… 97, 98.$
To find the number of whole numbers between $0$ and $99,$
Subtract $1$ from the difference of $0$ and $99.$
Therefore, Number of whole numbers between $0$ and $99 = (99 - 0) - 1$
$= 99 - 1$
$= 98$
Hence, the correct option is $(d).$

Positive number are located above 0 and negative numbers are located below $0$ on a vertical number line.

An additive identity is a number which when added to any number, the number is unchanged.
For example: $0 + 7 = 7, 0 + 9 = 9, 0 + (- 11) = -11$ etc.
Thus, $0$ is the additive identity in the set of whole numbers.
Hence, the correct option is $(a).$

Using distributive formula ,
We can write, $a(b + c) = a \times b + a \times c$
So, $36(8 + 3) = (36 \times 8) + (36 \times 3)$

Here, $x + 12 = 12 + 7.$
But, by commutative of addition,
We have $7 + 12 = 12 + 7$
So, comparing $x + 12 = 12 + 7$ and $7 + 12 = 12 + 7$
We have $x = 7$
Hence, the correct option is $(b).$

The numbers $1, 2, 3, 4, 5, ...$ form the collection of natural numbers.
Thus, $0$ is not a natural number.
Hence, the correct option is $(c).$

$[5 \times (6)] + [5 \times 3] = 5[(6) + 3] (\therefore$ distributive property$)$
$=5[9]$
$=45.$

Zero is called the identity element for addition of rational numbers.
Thus, the sum of $0$ and any rational number is the number itself.

Given Coordinate of Point $P$ are $(0, -3)$
$x - $coordinate $= 0$
$y -$ coordinate $= -3$
$\Rightarrow$ Point $P$ lies of $y -$ axis
Also, As $y -$ coordinate $ < 0$ Point $P (0, -3)$ lies on Negative $y -$ axis.

No. of whole numbers in between
$= (487 - 437) - 1 = 50 -1 = 49$

$\frac{-8}{17}+\text{x}=\frac{-8}{17}​\Rightarrow\text{x}=0$

The successor and predecessor of an odd natural number are both even.
Here, $n = o$ dd. So
Successor of $n = n + 1 ($Even natural number$)$
Predecessor of $n = n - 1 ($Even natural number$)$
Product $= (n + 1) \times (n - 1) =$ even $\times $ even $=$ even
Thus, the product of the successor and predecessor of $n$ is an even number.
Hence, the correct option is $(b).$

$ 16 - 2 ÷ 7 + 6 \times 2$
$=16-\Big(\frac{2}{7}\Big)+6\times2$ [Performing division]
$=16-\Big(\frac{2}{7}\Big)+12$ [performing multiplication]
$=25-\Big(\frac{2}{7}\Big)$ [Performing addition]
$=\frac{28}{1}-\frac{2}{7}=\frac{(196-2)}{7}=\frac{194}{7}=27\frac{5}{7}$ [performing substractrion]

Multiplication is associative for whole numbers.
i.e. $a \times (b \times c) = (a \times b) \times c$
$\therefore 24 \times (6 \times 5) = (24 \times 6) \times 5.$

We have:
$1$ million $= 10,00,000$
Predecessor of $1$ million,
$= 10,00,000 + 1 = 10,00,001$

$= 54 ÷ 3 of 6 + 9$
$= 54 ÷ (3 ÷ 6) + 9$
$= 54 ÷ 18 + 9$
$= 3 + 9 = 12$

Distributive property can bre written as
$a \times (b + c) = a\times b + a \times c$

$a + (b + c) = (a + b) + c$ is associative property of whole numbers.

Given expression:
$32-\Big[48\div\big[36-\big(27-\overline{16-9}\big)\big]\Big]$
$= 32 - [48 ÷ \{36 - (27 - 7)\}] [$Removing bar$]$
$= 32 - [48 ÷ \{36 - 20\}] [$Removing parentheses$]$
$= 32 - [48 ÷ 16] [$Removing braces$]$
$= 32 - 3 [$Removing square brackes$]$
$= 29$

The additive inverse of $36$ is $-36.$
i.e., $36 + (-36) = 0$

Multiplication is associative for whole numbers.
i.e. $a \times (b \times c) = (a \times b) \times c$
$\therefore 25 \times (15 \times 10) = (25 \times 15) \times 10$
$\therefore $ The missing terms are $10, 25.$

Commutative property of addition: Changing the order of addends does not change the sum.
For example, $a + b = b + a$

Multiplication is associative for whole numbers.
i.e. $a \times (b \times c) = (a \times b) \times c$
$\therefore 14 \times (2 \times 10) = (14 \times 2) \times 10$
$\therefore $ The missing terms are $2, 10.$

Divisor $= 46$
Quotient $= 11$
Remainder $= 15$
Dividend $=$ Divisor $\times $ quotient $+$ remainder
$= 46 \times 11 + 15$
$= 506 + 15$
$= 521$

Option $D$ is right.
Number of whole numbers lying between $-5$ and $5$ are $0, 1, 2, 3$ and $4.$

$(147 + 0) + (123 + 0) = 147 + 123$
$= 270$
$( \therefore 147 + 0 = 147, 123 + 0 = 123)$

The counting numbers $1, 2, 3, 4, .....$ are called natural numbers.
The set of natural numbers does not include $0.$

Smallest $3-$digit number $= 100$
Predecessor of $3-$digit number,
$= 100 - 1 = 99$
Hence, the correct option is $(b).$

The number $′0′$ along with the natural numbers gives us the number set $\{0, 1, 2, 3, 4, .....\}$ which is known as set of whole numbers.
Hence, $0, 1, 2, 3, 4....$ are whole numbers.

The smallest even whole number is $2.$
Zero is neither an even number nor an odd number.

Definition of multiplicative identity : an identity element $($such as $1$ in the group of rational numbers without $0)$ that in a given mathematical system leaves unchanged any element by which it is multiplied.
If we multiply $1$ with any number we will get the same number.

The set of whole numbers is $\{0, 1, 2, 3, 4, …\}$
So, the smallest whole number is $0$
Hence, the correct option is $(c).$

The smallest natural number is $1$ and the smallest whole number is $0.$
$\therefore 1 + 0 = 1$

A set of numbers which do not have any other common factor other than $1$ are called co-prime.
The $LCM$ of two co-prime numbers is equal to their product.

$= 13 - (12 - 6 ÷ 3)$
$= 13 - (12 - 2)$
$= 13 - 10 = 3$

 $\dfrac{-18}{5} = -3.6$
Therefore, it lies between $-3$ and $-4.$

 A multiplicative identity is a number which when multiplied with any number, the number is unchanged.
For example: $1 \times 7 = 7, 1 \times 9 = 9, 1 \times 100 = 100$ etc.
Thus, $1$ is multiplicative identity in the set of whole numbers is $1.$
Hence, the correct option is $(c).$

 Given expression:
$27-\Big[18-\big[16-\big(5-\overline{4-1}\big]\Big]$
$= 27 - [18 - \{16 - (5 - 3)\}] [$Removing bar$]$
$= 27 - [18 - \{16 - 2\}] [$Removing parentheses$]$
$= 27 - [18 - 14] [$Removing braces$]$
$= 27 - 4 [$Removing square brackes$]$
$= 23$

 The product of two odd numbers is an odd number.
Example: $5 \times 3 = 15$

$ 80808080 - 1 = 80808079$

Since, $99 = 100 - 1$
Therefore, $47 \times 99 = 47 \times (100 - 1)$
$= 47 \times 100 - 47$
$= 4700 - 47$
$= 4653$
Thus, the value of $47 \times 99$ is $4653.$
Hence, the correct option is $(b).$

 Here, $n$ is an even number.
Successor of $n = n + 1$
Predecessor of $n = n − 1$
Product $= (n + 1) \times (n − 1) = odd \times odd = odd$
Thus, the product of the successor and the predecessor of n is an odd natural number.
Hence, the correct option is $(b).$

$ a + (b + c) = (a + b) + c$ is associative property of whole numbers.

 The sum of $0$ and any rational number is the number itself.
Thus, for any rational number $\frac {\text{a}}{\text{b}}$​
we have $\frac {\text{a}}{\text{b}} + 0 = \frac {\text{a}}{\text{b}}​$

 In the product of two numbers, if one of the numbers (or both) is (are) zero, then the product is zero.
$\therefore0\times0=0$
In a division, if the numerator is zero, then the answer is zero.
$\therefore\frac{0}{2}=0$
Similarly, $\frac{(6-6)}{2}=\frac{0}{2}=0$
Here, $0$ is an additive identity in the set of whole numbers.
$\therefore4 + 0 =4$
Thus, $4 + 0$ is not zero.
Hence, the correct option is $(d).$

Here, $(31 + 15) + x = 31 + (15 + 23).$
But, by associativity of addition,
We have $(31 + 15) + 23 = 31 + (15 + 23)$
So, comparing $(31 + 15) + x = 31 + (15 + 23)$ and $(31 + 15) + 23 = 31 + (15 + 23),$
We have $x = 23$
Hence, the correct option is $(c).$

A set of numbers which do not have any other common factor other than $1$ are called co-prime.

Given expression:
$= 8 - [28 ÷ \{34 - (36 - 18 ÷ 9 × 8)\}] [$Performing division$]$
$= 8 - [28 ÷\{34 - (36 - 2 × 8)\}] [$Performing multiplication$]$
$= 8 - [28 ÷ \{34 - (36 - 16)\}]$
$= 8 - [28 ÷ \{34 - 20\}] [$Removing parentheses$]$
$= 8 - [28 ÷ 14] [$Removing braces$]$
$= 8 - 2$
$= 6 [$Removing square brackets$]$

We have:
Whole number $= 1$
Successor of $1 = 1 + 1 = 2$
Their product $= 1 \times 2 = 2$
Thus, $2$ is an even number.

Let $7589 - x = 3434$
Then, $x = 7589 - 3434 = 4155$

Given expression:
$= 100 \times 10 - 100 + 2000 ÷ 100$
$= 100 \times 10 - 100 + 20 [$performing division$]$
$= 1000 - 100 + 20 [$performing multiplication$]$
$= 1020 - 100 [$performing addition$]$
$= 920 [$performing subtraction$]$

The smallest prime number is $2.$
Thus, when we multiply any natural number we will always get an even number.

Commutative property of addition: Changing the order of addends does not change the sum.
For example, $a + b = b + a$

The whole numbers between $437$ and $487$ are $438, 439, 440, 441, … , 484, 485$ and $486.$ To find the required number of whole numbers,
We need to subtract $437$ from $487$ and then subtract again $1$ from the result.
Thus, there are $(487 - 437) - 1$ whole numbers between $437$ and $487.$
Now, $(487 - 437) - 1$
$= 50 - 1 = 49$
Hence, the correct option is $(b).$

Predecessor of $1 = 1 - 1 = 0$

The predecessor of an odd number is an even number.
The successor of an odd number is also an even number.
These two even numbers are two consecutive even numbers, and the product of two consecutive even numbers is always divisible by $8.$

The additive identity property says that if you add a real number to zero or add zero to a real number, then you get the same real number back.
The number zero is known as the identity element.
Then zero $(0)$ is the additive identity of a real number and all rational numbers are real.
Hence, $0$ is the additive identity of rational numbers.

We have:
Successor of $99 = 99 + 1 = 100$
Predecessor of $99 = 99 - 1 = 98$
Their product $= 100 \times 98 = 9800$

Therefore, the identity element for addition of whole numbers is $0.$

 By using the distributive property, we have;
$1568 \times 185 - 1568 \times 85$
$= 1568 \times (185 - 85)$
$= 1568 \times 100$
$= 156800$

$ 5$ and $8$ are whole numbers but $-3$ is not a whole number.

 Apart from $B,$ the other options contain a negative sign.
Negative numbers do not belong to whole numbers.

The whole number which has no predecessor is $0.$
i.e., $0 - 1 = -1,$ which is not a whole number.

Let the number is $n$
Then the successor of the number $= n + 1$
and the predecessor of the number $= n − 1$
Difference $= n+1−(n−1)$
$= n + 1 − n + 1$
$= 2$

Here we have to tell what least number should be subtracted from $10004$ to get a number exactly divisible by $12.$
So, we will first divide $10004$ by $12.$

Remainder $= 8$
So, $8$ should be subtracted from $10004$ to get the number exactly divisible by $12.$
i.e., $10004 - 8 = 9996$​​​​​​​

Hence, $9996$ is exactly divisible by $12.$

The distributive property of multiplication states that a number is multiplied by the sum of two numbers, when the first number can be distributed to both of those numbers and multiplied by each of them separately, then adding the two products together for the same result as multiplying the first number by the sum.
So $(4 \times 8) + (4 \times 60) = 4 \times (8 + 60)$

Commutative property for addition holds true in the case of whole numbers.
For example:
$7 + 4 = 11$
$4 + 7 = 11$
$\Rightarrow 7 + 4 = 4 + 7$

$(18 + 59) + 1 = 18 + (59 + 1) ($associative property$)$
$= 18 + 60$
$= 78$
$($Here, to find the result, we avoided $18 + 59,$ which could be time consuming$).$

Smallest whole number is $0$ and natural number is $1$

Number $-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5$ lies between $-6$ and $6.$
Among them $0, 1, 2, 3, 4, 5$ are whole numbers.
$\therefore 6$ whole numbers lies between $-6$ and $6$

The multiplicative identity property states that any time you multiply a number by $1,$ the result, or product, is that original number.
So it is clear that multiplicative property is given by

Addition of whole numbers is commutative.
So, for any two whole numbers, $x$ and $y, x + y = y + x$

Even natural number $= 2$
Predecessor of $2 = 2 - 1 = 1$
Successor of $2 = 2 + 1 = 3$
Their product $= 1 \times 3 = 3$
Thus, the product is an odd number.

 Addition is associative for whole numbers
i.e. $a + (b + c) = (a + b) + c$
$\therefore 571 + (429 + 394) = (571 + 429) + 394$ 
$\therefore$ The missing terms are $429, 571.$

$\frac{-4}{7}+\frac{0}{9}=\frac{-4}{7}$
$\frac{-4}{7}-\frac{0}{9}=\frac{-4}{7}$
Therefore we use both $+$ or $-$

Since in succession we move from left to the right on the number line.
And , $-17$ lies on right side of $-18$
Hence, $-17$ is the successor of $-18$

 The smallest whole number is $0.$
Greatest three digit number $= 999$
The number of whole numbers from $0$ to $999$ is $1000.$
So, the number of whole numbers between $0$ and $999 ($excluding $0$ and $999)$ is $1000 - 2 = 998.$
Hence, the correct option is $(c).$

$ (888 + 777 + 555) = (111 \times ?)$
$\Rightarrow (888 + 777 + 555) = 111 \times (8 + 7 + 5) [$by taking $111$ common$]$
$= 111 \times 20 = 2220$

Here, Divisor $= 53,$ Quotient $= 8$ and Remainder $= 5.$
Now, using the relation Dividend $=$ Divisor $\times $ Quotient $+$ Remainder
We get Dividend,
$= 53 \times 8 + 5$
$= 424 + 5$
$= 429$
Thus, the required number is $429.$
Hence, the correct option is $(c).$

 $\frac{11}{4}=2\frac{3}{4}$
So $\frac{11}{4}$ lies between $2$ and $3.$

Addition is associative for whole numbers. i.e. $a + (b + c) = (a + b) + c$
$\therefore  52 + (53 + 55) = (52 + 53) + 55$
$\therefore$ The missing terms are $52, 53.$

$ 1, 2, 3, 4, 5, .....$ are called "natural numbers" and are also known as counting numbers.

By regrouping, we get
$4 \times 378 \times 25 = 4 \times 25 \times 378$
$= 100 \times 378$
$= 37800$
Hence, the correct option is $(a).$

If two numbers are equal, then their $LCM$ is equal to their $HCF.$

 Successor of $999 = 999 + 1 = 1000$
Predecessor of $1001 = 1001 - 1 = 1000$
Now,
Product$ = ($Successor of $999) × ($Predecessor of $1001)$
$= 1000 × 1000$
$= 1000000$
$=$ One million
Hence, the correct option is $ (c).$

The sum of any two whole numbers and the product of the same are not necessarily equal.
Eg. $2 + 2 = 4 = 2 \times 2,$ but $3 + 3 = 6$ and $3 \times 3 = 9$ which are not equal.
Rest all options depicts commutative property which whole numbers satisfy.

Since, factors of,
$91 = 1 \times 7 \times 13$
$81 = 1 \times 3 \times 3 \times 3 \times 3$
$87 = 1 \times 3 \times 29$
$97 = 1 \times 97$
Thus, $81, 87$ and $91$ all are not prime numbers.

Product of two numbers $= HCF$ of two numbers $\times \ LCM$ of two numbers.
$\Rightarrow 1530 = 15 \times LCM$ of two numbers
$\Rightarrow LCM$ of two numbers $= 153015 = 102$
$\Rightarrow 1530 = 15 \times LCM$ of two numbers
$\Rightarrow LCM$ of two numbers $= 153015 = 102$

Here, $n + 35 = 101.$
Adding $-\ 35$ on both sides, we get
$n + 35 + (-35) = 101 + (-35)$
$n + 0 = 66$
$n = 66$
Hence, the correct option is $(d).$

Natural numbers start at $1$ and as we add $1$ to a natural number, we get the next natural number.
So if we keep adding $1,$ we keep getting next natural numbers, so the natural numbers never ends.

Successor of $1001 = 1001 + 1 = 1002$

Given expression:
$= 3640 - 14 ÷ 7 \times 2$
$= 3640 - 2 \times 2 [$performing division$]$
$= 3640 - 4 [$performing multiplication$]$
$= 3636 [$performing subtraction$]$

We have:
$15 + 5 ÷ 5 \times 2$
$= 15 + 1 \times 2 ($Performing division$)$
$= 15 + 2 ($Performing multiplication$)$
$= 17$

$1$ is the only number which when multiplied to any number $82,$ gives the same number.

Associative property for addition states that for any three numbers,
it does not matter which two are added first and then their sum is added to the third integer the final result will be same$\therefore a + (b + c) = (a + b) + c$

Here, we have to tell that what least number must be added to $10056$ to get a number exactly divisible by $23.$
So, first we will divide $10056$ by $23.$

Remainder $= 5$
Required number $= 23 - 5 = 18$
So, $18$ must be added to $10056$ to get a number exactly divisible by $23.$
i.e., $10056 + 18 = 10074$​​​​​​​

Hence, $10074$ is exactly divisible by $23.$

$LHS = 0 + (8 + 5) = 0+ 13 = 13$
$RHS = (0 + 8) + 5 = (0 + 8) + 5 = 13$
$LHS = RHS$