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Exercise

Problems on Numbers

Direction To Solve

Choose the correct option

1- 186 x 186 + 159 x 159 - 2 x 186 x 159 =?

A.329
B.700
C.729
D.848
E.None of these

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2- The number of prime factors of (3 x 5)¹² (2 x 7)¹⁰ (10)²⁵ is:

A.69
B.84
C.93
D.94
E.None of these

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3- If (64)² - (36)² = 20z, the value of z is:

A.140
B.142
C.684
D.670
E.None of these

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4- Which of the following numbers is divisible by 3, 7, 9 and 11?

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5- 39798 + 3798 + 378 =?

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6- A six-digit number is formed by repeating a three-digit number; for example, 256256 or 678678 etc. Any number of this form is always exactly divisible by:

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7- The unit's digit in the product (3127)¹⁷³ is:

A.1
B.3
C.7
D.9
E.None of these

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8- The number of digits of the smallest number, which when multiplied by 7 gives the result consisting entirely of nines, is:

A.3
B.5
C.6
D.8
E.None of these

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9- Which of the following numbers is exactly divisible by 99?

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10- On dividing a number by 999, the quotient is 366 and the remainder is 103. The number is:

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Imoprtant Tips

IMPORTANT FACTS AND FORMULAE

I. Numeral : In Hindu Arabic system, we use ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number.

A group of digits, denoting a number is called a numeral.

We represent a number, say 689745132 as shown below :

Ten Crores (10⁸)	Crores (10⁷)	Ten Lacs (Millions) (10⁶)	Lacs (10⁵)	Ten Thousands (10⁴)	Thousands (10³)	Hundreds (10²)	Tens (10¹)	Units (10⁰)
6	8	9	7	4	5	1	3	2

We read it as: 'Sixty-eight crores, ninety-seven lacs, forty-five thousand, one hundred and thirty-two'.

II. Place Value or Local Value of a Digit in a Numeral :

In the above numeral :
Place value of 2 is (2 x 1) = 2; Place value of 3 is (3 x 10) = 30;
Place value of 1 is (1 x 100) = 100 and so on.
Place value of 6 is 6 x 10⁸ = 600000000

III. Face Value : The face value of a digit in a numeral is the value of the digit itself at whatever place it may be. In the above numeral, the face value of 2 is 2; the face value of 3 is 3 and so on.

IV. TYPES OF NUMBERS

1. Natural Numbers: Counting numbers 1, 2, 3, 4, 5,..... are called natural numbers.

2. Whole Numbers: All counting numbers together with zero form the set of whole numbers.
Thus,
(i) 0 is the only whole number which is not a natural number.
(ii) Every natural number is a whole number.

3. Integers : All natural numbers, 0 and negatives of counting numbers i.e.,
{…, - 3 , - 2 , - 1 , 0, 1, 2, 3,…..} together form the set of integers.
(i) Positive Integers: {1, 2, 3, 4, …..} is the set of all positive integers.
(ii) Negative Integers: {- 1, - 2, - 3,…..} is the set of all negative integers.
(iii) Non-Positive and Non-Negative Integers : 0 is neither positive nor negative. So, {0, 1, 2, 3,….} represents the set of non-negative integers, while {0, - 1 , - 2 , - 3 ,…..} represents the set of non-positive integers.

4. Even Numbers: A number divisible by 2 is called an even number, e.g., 2, 4, 6, 8, 10, etc.

5. Odd Numbers: A number not divisible by 2 is called an odd number. e.g., 1, 3, 5, 7, 9, 11, etc.

6. Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.
Prime numbers upto 100 are : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
Prime numbers Greater than 100: Letp be a given number greater than 100. To find out whether it is prime or not, we use the following method:
Find a whole number nearly greater than the square root of p. Let k > *jp. Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime.
e.g,,We have to find whether 191 is a prime number or not. Now, 14 > V191.
Prime numbers less than 14 are 2, 3, 5, 7, 11, 13.
191 is not divisible by any of them. So, 191 is a prime number.

7. Composite Numbers : Numbers greater than 1 which are not prime, are known as composite
numbers, e.g., 4, 6, 8, 9, 10, 12.
Note : (i) 1 is neither prime nor composite.
(ii) 2 is the only even number which is prime.
(iii) There are 25 prime numbers between 1 and 100.

8. Co-primes: Two numbers a and b are said to be co-primes, if their H.C.F. is 1. e.g., (2, 3), (4, 5), (7, 9), (8, 11), etc. are co-primes,