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 4a = 5, 5b = 6, 6c = 7 and 7d = 8 then (a x b x c x d) = ?
Explanation:
8 = 7d = (6c)d = (5b)cd = (4a)bcd = 22abcd
8 = 23 = 22abcd
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 (n3 – n) ?
Explanation:
(n3 – n) = n (n – 1) (n + 1)
n = 1, (n3 – n) = 0;
n = 2, (n3 – n) = 2 x 1 x 3 = 6;
n = 3, (n3 – n) = 3 x 2 x 4 = 6 x 4;
n = 4, (n3 – n) = 4 x 3 x 5 = 6 x 10;
n = 5, (n3 – n) = 5 x 4 x 6 = 6 x 20;
Hence, (n3 – n) is always divisible by 6.
Q:- For n = Natural Number, Find the number to divide (34n – 43n) always?
Explanation:
(34n – 43n) = (34)n – (43)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:
(xn + yn) 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, (xn -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, (xn +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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