Wednesday, September 13, 2023

FRIENDLY NUMBER

                                                                      FRIENDLY NUMBERS

FRIENDLY NUMBER is different from Friend / Amicable Numbers. So don't get confused with its similarity.

Friendly Number

Two or more natural numbers are called friendly numbers if they have common abundancy index. The common abundancy index is decided by dividing the sum of all divisors of a number including itself by the number itself.

Example:- 30 has the divisors – 1, 2, 3, 5, 6, 10, 15 and 30

The sum of divisors = 1 + 2 + 3 + 5 + 6 + 10 + 15 + 30 = 72

Abundancy index = 72/30 = 12/5

The abundancy index of a number could be a natural number or rational number.

If the sum of divisors of a number  (excluding itself) = Number itself , it is called Perfect Number

If the sum of divisors of a number  (excluding itself) >Number itself , it is called Abundant Number

If the sum of divisors of a number  (excluding itself) < Number itself , it is called Deficient Number

6 and 28 are Perfect Numbers

Divisor of 6 = 1, 2 and 3 . Sum of divisors = 1 + 2 + 3 = 6

Divisors of 28 are = 1, 2, 4 ,7 and 14 . Sum of divisors = 1 + 2 + 4 + 7 + 14 = 28

6 and 28 are friendly number with abundancy index 2, i.e. any number with abundancy index 2 is called Friendly numbers.

Divisors of 6 = 1, 2, 3 and 6

Sum of Divisors = 1 + 2 + 3 + 6 = 12

Abundancy Index = 12 /6 = 2

Divisor of 28 = 1, 2, 4, 7, 14 and 28

Sum of divisors = 1 + 2 + 4 + 7 + 14 + 28 = 56

Abundancy Index = 56 / 28 = 2

Since 6 and 28 have same abundancy index so they are FRIENDLY NUMBERS.

More examples of friendly numbers are --- (30,140) (2480, 6200 and 40640)

 If you want to know about Amicable/Friends Number 


FRIEND NUMBER/ AMICABLE NUMBERS

READ ANOTHER ARTICLE 

Numbers with Cool Names: Amicable, Sociable, Friendly – TOM ROCKS MATHS


LIST OF DIFFERENT NUMBERS

Do write your comment in comment box.


DR RAJESH KUMAR THAKUR


 

 

 

Saturday, December 12, 2020

Finding Remainder when Dividend is in power





Dr Rajesh Kr Thakur

Finding Remainder Easy Part - 1

 

Finding Remainder

Do you think finding remainder is easy?

Yes, if you are dividing one big number by smaller.

But consider a situation where you find yourself perplexed and helpless and you don’t have any idea to calculate the remainder because you can’t apply the basic rule –

Dividend = Divisor × Quotient + Remainder

Let’s first begin with Euclid Division lemma

As you know that Euclid is called the father of geometry but he has contributed immensely in Arithmetic and Number theory also.

Euclid Division Lemma

Let a and b be any two positive integers then there exist unique integers q and r such that a = bq + r where 0 ≤ r < b

This is basically the Euclid Division Lemma

                          

In the first example Dividend = 445, Divisor = 17,   Quotient = 26 and Remainder = 3

Apply the division lemma

Dividend = Divisor × Quotient + Remainder

          445 = 17 × 26 + 3

                 = 442 + 3

Here, Remainder < Divisor

In the second example, remainder = 0 which also satisfies the division lemma. Let’s check

Dividend = Divisor × Quotient + Remainder

2412 = 36 × 67 + 0

2412 = 2412

But what will be the remainder when 2 is divided by 6?

Remember, if numerator is less than denominator, i.e. when dividend is less than divisor then dividend will be remainder.

Here, obviously, 2 is the remainder

Some people write 2 ÷ 6 as and further cancel out to get and write 1 as quotient which is wrong. Finding remainder is something different from simplifying fraction or changing a fraction into simplest form. Therefore, you are advised not to cancel the number.

In the above two examples we have seen the remainder is a positive integer but in finding the remainder of larger number the concept of negative integer remainder also plays a big role.

Negative Remainder

Let’s take an example

26 = 9 × 2 + 8

26 = 9 × 3 – 1

In the first case remainder = + 8 while in the second case remainder = –1.

Negative remainder reduces the effort of calculation.

54 = 11 × 4 + 10                                             Remainder = + 10

54 = 11 × 5 – 1                                                Remainder = – 1

 

Finding remainder of product

Example: - Find the remainder when 33 x 34 x 35 is divided by 8

Solution: - The first thing we can do it to multiply 33, 34 and 35

                        33 × 34 × 35 = 39270

Now divide 39270 by 8 . What do you get?

Remainder = 6

Let’s try it by remainder method.

On division of 33 by 8 we get remainder = 1

On division of 34 by 8 we get remainder = 2

On division of 35 by 8 we get remainder = 3



Hence  Remaider = 6

The conclusion that we can draw from the above example is that if a, b, c ---- be numbers divided by w gives remainder x, y, z - - respectively then the product of a, b, c - - - when divided by the same divisor w will give the same remainder obtained by dividing the product of remainder x, y, z --- by w.




First divide 121, 118 and 124 by 60 to get remainders as 1, –2 and 4 respectively.

Therefore final remainder can be obtained without actual multiplication of 121× 118 × 124 and it will be the product of individual remainder of numbers and i.e.  (1×–2×4) = –8

Final remainder = –8 + 60 = 52.

Example: Find the remainder when 37500 is divided by 9

Solution:- As we know that 34 = 3 × 3 × 3 × 3 = 81

Hence, 37500 = 37 × 37 × - - - 500 times.

Multiplying 37 five hundred times is a big deal so we need to think something extra ordinary. We divide 37 by 9 and get the remainder 1

Therefore, 37500  divided by 9 is equal to 1500divided by 9

Hence, Remainder = 1

Example: Find the remainder when 54 × 63 × 86 is divided by 11

Solution:-

On division of 54 by 11 we get remainder = 10

On division of 63 by 11 we get remainder = 8

On division of 86 by 11 we get remainder = 9

Through mental calculation we can find the remainder on dividing 720 by 11.

720 = 11 × 65 + 5; Remainder = 5

Let’s apply the concept of negative integer

On division of 54 by 11 we get remainder = – 1 because 54 – 5 × 11 = – 1

On division of 63 by 11 we get remainder = – 3 because 63 – 6 × 11 = –3

On division of 86 by 11 we get remainder = – 2 because 86 – 8 × 11 = – 2

Since final remainder can’t be – 6, so we add the divisor 11 to make the remainder positive at the end.

Remainder = – 6 + 11 = 5

This shows that the negative remainder concept saves our time.

DR Rajesh Thakur


Wednesday, December 9, 2020

यूनिट टेस्ट क्लास 6 / UNIT TEST CLASS 6

 

प्यारे बच्चों

यूनिट टेस्ट 20 नम्बर का है और इसे 30 मिनट में करने हैं .

सबसे पहले टेस्ट के प्रश्न करो और उत्तर लिख कर रखो . जब टेस्ट हो जाये तो सबसे नीचे टेस्ट का link दिया है उसे क्लिक करो और फिर टेस्ट का उत्तर भरो. साथ ही अपन नाम और रोलनम्बर लिखो 

1. The product of  (-7/8) and 4/21 is   / (-7/8) और   4/21का गुणनफल है ?

a)-1/6  b)12 c) -63/16 d) -16/147

2. If 2XY7 is a four digit number divisible by 3 then the least value of X + Y  is /  2X Y7 चार अंक की एक संख्या है जो 3 से विभाजित है तो x + y का न्यूनतम मान क्या होगा

a)3   b)4 (c) 6 (d) 6

3. The fraction representing the shaded portion is/ दिए चित्र में छायांकित भाग का हिस्सा है 




(a) 
1/4
(b) 1/2
(c) 1/8
(d) none of these

4. How many diagonals are there in the following figure? / दिए चित्र में कितने विकर्ण हैं 




(a) 4
(b) 5
(c) 2
(d) 3

5. The difference between the successor of a number and the number it self is – किसी संख्या की अगली संख्या और संख्या के बीच का अंतर है
(a) 0
(b) – 1
(c) 1
(d) none of these.

6.The smallest of the numbers1000,50000,111,3222,225 is / इनमे से सबसे छोटी संख्या है

(a) 111
(b) 225
(c) 1000
(d) 3222

7. Find the value of x in 7x + 15 = 50 / यदि 7x + 15 = 50 हो तो x का मान होगा

a) 15         b) 4          c) 5          d) 0

8. If my age 10 years ago was x then what will be my age 10 years from now. / यदि 10 वर्ष पहले मेरी उम्र x वर्ष है तो 10 वर्ष के बाद मेरी उम्र होगी

a) 10         b) 20         c) x + 10            d) x + 20

9. 10 + 2 + 1/10 + 2/100 =
(a) 12.12
(b) 12.21
(c) 11.11
(d) 21.12

10. Standard form of 0.00001275 is / 0.00001275 का मानक रूप होगा ?

          a) 1.275 * 10-5         b). 1.275 * 105        c). 127.5 * 10-7        d). 127.5 * 107

टेस्ट करने के बाद इस link को क्लिक करो

डॉ राजेश ठाकुर 

 

November 2, 2024