**What is Multiple?**

All the number of a table of any number is called the multiple.

Example:

4 x 1 = 4

4 x 2 = 8

4 x 3 = 12

…………

…………

4, 8, 12,……. are the multiple of 4.

**Note:- 0 is the multiple of any number.**

**What is Factor?**

One or more than one number divide another number. The one or more than one number is known as the factor.

Example:

12 ÷ 1 = 12

12 ÷ 2 = 6

12 ÷ 3 = 4

12 ÷ 4 = 3

12 ÷ 6 = 2

12 ÷ 12 = 1

1, 2, 3, 4, 6 and 12 are the factor of 12.

**Note:** Every number has minimum two factors. Like 1 and itself.

**What is LCM (Lowest Common Multiple)?**

2 x 2 x 2 x 3 x 2 x 3 = 144

The least number 144 is completely divisible by 8, 12, 16 and 18. The number 144 is the Lowest Common Multiple (LCM) of 8, 12, 16 and 18.

**What is HCF (Highest Common Factor)?**

HCF of 18 and 24 = 6

One of the largest numbers 6 divides the number 18 and 24. The HCF of 18 and 24 is 6.

Explanation:

18 ÷ 1 = 18 24 ÷ 1 =24

18 ÷ 2 = 9 24 ÷ 2 = 12

18 ÷ 3 = 6 24 ÷ 3 = 8

18 ÷ 6 = 3 24 ÷ 4 = 6

18 ÷ 9 = 2 24 ÷ 6 = 4

18 ÷ 18 = 1 24 ÷ 12 = 2

24 ÷24 = 1

The factors of 18 are 1, 2, 3, ** 6**, 9, 18.

The factors of 24 are 1, 2, 3, 4, ** 6**, 12, 24.

The Highest Common Factor (HCF) of 18 and 24 is 6.

**Formula of HCF and LCM**

∝:- The Product of two number = HCF of two number x LCM of two number

∝:- HCF of Fraction = HCF of Numerator/LCM of Denominator

∝:- LCM of Fraction = LCM of Numerator/HCF of Denominator

**How to find the least number of the following numbers?**

3/5, 5/7, 3/7, 5/9, 7/9

**Explanation:-**

We change the following numbers are in decimal form.

3/5 = 0.60

5/7 = 0.71

3/7 = 0.42

5/9 = 0.55

7/9 = 0.77

Hence, The least number is 3/7.

**How to find the largest number of the following numbers?**

3/5, 5/7, 3/7, 5/9, 7/9

**Explanation:-**

We change the following numbers are in decimal form.

3/5 = 0.60

5/7 = 0.71

3/7 = 0.42

5/9 = 0.55

7/9 = 0.77

Hence, The largest number is 7/9.

**How to arrange the following numbers in increasing order?**

3/5, 5/7, 3/7, 5/9, 7/9

**Explanation:-**

We change the following numbers are in decimal form.

3/5 = 0.60

5/7 = 0.71

3/7 = 0.42

5/9 = 0.55

7/9 = 0.77

Hence, Increasing Order: 3/7<5/9<3/5<5/7<7/9

**How to arrange the following numbers in decreasing order?**

3/5, 5/7, 3/7, 5/9, 7/9

**Explanation:-**

We change the following numbers are in decimal form.

3/5 = 0.60

5/7 = 0.71

3/7 = 0.42

5/9 = 0.55

7/9 = 0.77

Hence, Decreasing Order: 7/9>5/7>3/5>5/9>3/7

**Technique for Problems**

Q:- **If 4**^{a}** = 5, 5**^{b}** = 6, 6**^{c}** = 7 and 7**^{d}** = 8 then (a x b x c x d) = ?**

**Explanation:**

8 = 7^{d} = (6^{c})^{d} = (5^{b})^{cd} = (4^{a})^{bcd} = 2^{2abcd}

8 = 2^{3} = 2^{2abcd}

3 = 2abcd

Abcd = 1.5 Ans.

Q:- **If (2/5) ^{6 }(2/5)^{-8} = (2/5)^{3x-8} then, x = ?**

**Explanation:**

(2/5)^{6 }(2/5)^{-8} = (2/5)^{3x-8}

(2/5)^{6-8} = (2/5)^{3x-8}

(2/5)^{-2} = (2/5)^{3x-8}

3x – 8 = – 2

3x = 6

x = 2 Ans.

Q:- **For n = Natural Number, find the greatest number to divide (n ^{3} – n) ?**

**Explanation:**

(n^{3} – n) = n (n – 1) (n + 1)

n = 1, (n^{3} – n) = 0;

n = 2, (n^{3} – n) = 2 x 1 x 3 = 6;

n = 3, (n^{3} – n) = 3 x 2 x 4 = 6 x 4;

n = 4, (n^{3} – n) = 4 x 3 x 5 = 6 x 10;

n = 5, (n^{3} – n) = 5 x 4 x 6 = 6 x 20;

Hence, (n^{3} – n) is always divisible by 6.

Q:- **For n = Natural Number, Find the number to divide (3 ^{4n} – 4^{3n}) always?**

**Explanation:**

(3^{4n} – 4^{3n}) = (3^{4})^{n} – (4^{3})^{n} = (81)^{n} – (64)^{n}

So, (x)^{n} – (y)^{n} has always a divisibility of (x – y).

Hence, 81 – 64 = 17 Ans

Q:- **Find the common factor of {(123) ^{45} + ( 119)^{45}} and {(123)^{45} + ( 119)^{45}}?**

**Explanation:**

(x^{n} + y^{n}) has a factor (x + y)

So, 123 + 119 = 242

Hence, the common factor is 242 Ans.

Q:- **Find the number of zero in the right side of product of 1, 2, 3, 4, …….and 100?**

**Explanation:**

The Number of zero in the product of 1 to 10 = 2

The Number of zero in the product of 11 to 20 = 2

The Number of zero in the product of 21 to 30 = 2

…………………………………………………

The Number of zero in the product of 91 to 100 = 3

The total number of zero are 21.

Q:- **If (15) ^{100} is divided by 16. What will be reminder?**

**Explanation:**

If n is an even number, then, (x^{n} -1) is always divisible by (x + 1)

So, we can write:

(15)^{100} = [(15)^{100} – 1)] + 1

Hence, (15)^{100} is divisible by 16. We get reminder 1. Ans

Q:- **If (35) ^{15 }is divided by 36. What will be the reminder?**

**Explanation:**

If n is an odd number, then, (x^{n} +1) is always divisible by (x + 1)

So, we can write:

(35)^{15} = [(35)^{15} + 1)] – 1

Hence, (35)^{15} is divisible by 36. We get reminder = (36 – 1) = 35. Ans

Q:- **If two digit number repeat two times to make a four digit number like 1313, 1515. Find the number that divides the four digit number?**

**Explanation:**

Let the number be xyxy.

1000x +100y + 10x + y

= 100 (10x + y) + (10x + y)

= (100 + 1) (10x + y)

= 101 (10x + y)

Hence, the number is always divisible by 101. Ans.

Q:- **Find the least number whose divided by 12, 15 and 16. We get the reminder 7, 10 and 11 respectively?**

**Explanation:**

Here, (12 – 7) = (15 – 10) = (16 – 11) = 5

We can find the LCM of 12, 15 and 16 = 240

= 240 – 5 = 235 Ans.

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