A.) Basic Concepts:
Common things related to topic:
Natural Numbers: Orders or Counting numbers is known as Natural numbers. Example: 1,2,3,4,...
Whole Numbers: A positive integers starting from 0 is known as Whole numbers. Example: 0,1,2,3,4,...
Prime Numbers: A number which has no factor and or also which can't be divisible by any number is known as Prime numbers. Example: 2,3,5,7,11. Of all the numbers, 2 is the only even prime number.
Integer: An Integer is any number in the set. Example: -2,-1,0,1,2...
Even Numbers: A number which is exactly divisible by 2 is known as Even numbers. Even numbers is said to be number which is placed in unit digit. Example: 2,6,18,24....
Odd Numbers: A number which is not divisible by 2 is known as Odd numbers. Odd numbers is said to be number which is placed in unit digit. Example: 1,3,7,11,13...
Real Numbers: All rational and irrational number is called as Real Numbers.
Rational Numbers: A number which is not divided by 0 is called as Rational Numbers. Example: 1/8= 0.125, 2/24= 0.0833..
Irrational Numbers: A number which is divided by 0 or the result goes on infinity is called as Irrational numbers. Example: 1/0, √5,...
Reciprocal: It is inversing the division format, which when multiplies gives 1 as the answer. Example: 8/9 so (8/9) * (9/8)=1, as like that.
Least Common multiple(LCM): When two or more numbers, the least or smallest common number in the set of common multiples is known as LCM. Example: The common multiple for 2 and 3 is, whereas 2 is 2,4,6 and 3 is 3,6. So 6 is common number.
Absolute Value: The absolute value of a number is the equivalent positive value. Example: |-2| = 2 (or) |4| = 4.
The mathematical concept is done upon the four basic operations. Addition, 2+2=4; Subtraction, 7-4=3; Multiplication, 4*5=20; Division, 6/3=2.
Basic Formulas:
i.) (a + b)(a - b) = (a² - b²)
ii.) (a + b)² = (a² + b² + 2ab)
iii.) (a-b)² = (a² + b² - 2ab)
iv.) (a +b+ c)² = a² + b² + c² + 2(ab + bc+ ca)
v.) (a³ + b³) = (a + b)(a² - ab + b²)
vi.) (a³ - b³) = (a - b)(a² + ab + b²)
vii.) (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac)
viii.) Sum of first 'n' natural numbers is given by (n(n+1))/2.
ix.) Sum of first 'n' odd numbers is given by n². Sum of first 'n' even numbers is given by n(n+1).
B.) Divisibility:
Divisibility rule:
It is a rule, which is used to find the divisible number to avoid the long division process. Shortly to say is, to find the divisible number in shortcut.
2 ➤ If the number is even. Example: 2018, 2016, 4, 6.
3 ➤ If the sum of the digits is divisible by 3. Example: 2328 = 2+3+2+8 = 15, is divisible by 3; 3127 = 3+1+2+7 = 13, is not divisible by 3.
4 ➤ If the number formed by last two digits is divisible by 4. Example: 1528, 28 is divisible by 4. So, 1528 is divisible by 4.
5 ➤ Ends with 5 or 0. Example: 1225, 100, 125.
6 ➤ If the number is divisible by both 2 and 3. Example: 216, divisible by 2 as it is even, divisible by 3 as sum of digits is 2+1+6=9. So, 216 is divisible by 6.
7 ➤ Take the last digit and double it. Subtract from the remaining digits, continue until you get a number that gives a -7,7 or 0. Example: 1337, take last digit 7, double it 7*2=14. Now subtract from the remaining 133-14=119. Now, 119 same process, 9*2=18, 11-18=-7. So, 1337 is divisible by 7.
8 ➤ If the number formed by last three digits is divisible by 8. Example: 26128, 128 are divisible by 8. So, 26128 is divisible by 8.
9 ➤ If the sum of the digits is divisible by 9. Example: 1998, 1+9+9+8=27 is divisible by 9. So, 1998 is divisible by 9.
10 ➤ If the last digit of a number is 0. Example: 2000, 2020.
11 ➤ If (sum of digits at odd places - sum of digits at even place) = multiple of 0 or 11. Example: 121, (1+1)-(2)=0. Hence, 121 is divisible by 11.
12 ➤ A number which is divisible by both 3 and 4. Example: 108, 108/3=36 and 108/4=27. So, 108 is divisible by 12.
Last digit of a number Its Square Its Cube
0 0 0
1 1 1
2 4 8
3 9 7
4 6 4
5 5 5
6 6 6
7 9 3
8 4 2
9 1 9
Examples from the above content:
Example 1: Which of these numbers is divisible by 24?
a.12493 b.12137 c.32424 d.2526
Solution: Divisibility test should be 3x8 not 4x6(they have a common factor).
So from the given options, 12493 and 12137 can't be divisible because it is odd.
32424, sum of digits=3+2+4+2+4=15. Hence it is divisible by 3.
Number formed by last 3 digits =424, is a multiple of 8. So 32424 is divisible by 24.
**Likewise, if any two or more digits number is given to divisible, make it uncommon factor like this in the example.
Example 2: Find the remainder when 528528528.... up to 528 digits is divided by 27?
Solution: 528528528.... up to 528 digits mod 21 = 3 * (176176.... up to 528 digits mod 9) - Equation 1
The digital sum of 176176176... up to 528 digits = (1+7+6) * 176 = 2464 whose digital sum is 7 which is two short of 9.
Hence the remainder is 9-2 = 7.
So overall remainder using Equation 1 is = 3 * 7 = 21
Example 3: What is the remainder when 3^24 - 1 is divided by 80 ?
Solution: (3^24 - 1) mod 80 = 9^12 mod 80 - 1 mod 80
= 81^6 mod 80 - 1 mod 80
= (1-1) mod 80 = 0.
Example 4: What is the remainder when 6^17+17^6 is divided by 7?
Solution: 6^17 mod 7 = (-1)^17 mod 7, because 7 must be divided with the first number like 6/7 it gives a decimal value. So, we should subtract that number. Hence now it becomes (-1)^17 mod 7 = -1 mod 7(As 17 is odd so power becomes odd) = -1
Now, 17^6 mod 7, in this 17 must divide with 7. So, 17/7 = 3 remainder.
Now, 3^6 mod 7 = 729 mod 7 = 1.
So the remainder when 6^17+17^6 is divided by 7 is -1+1 = 0.
So the remainder is 0.
How to find MOD value?
Example 5: 20 MOD 7= ?
In this, we should divide 20/7 = 2.85 -> 2.
Just take the decimal value if it is 2.32 or 2.85 and put the number 2.
So, 20 - 2*7= 6. Hence, 20 MOD 7 = 6.
**Note: It is the shortcut for find remainder purposes.
Example 6: The remainder when integers a, b are divided by 7 are 5 and 4. What is the remainder when a*b is divided 7?
Solution: Using remainder theorem, a*b mod 7 = 5*4 mod 7 =20 mod 7 = 6.
Example 7: What is the difference between the largest and smallest number written with all the four digits 9,3,1 and 4?
Solution: Largest number written with 9,3,1 and 4=9431. Smallest number written with 1349, Difference = 9431-1349=8082.
Example 8: If 95 is multiplied by a certain number, and that number is increased by 7332. Find the number?
Solution: Let the number be n. nx95=n+7332.
94n=7332, n=78.
Example 9: What is the number of prime factors in the expression (6)^10*(17)^17*(19)^27?
Solution: Given, (6)^10*(17)^17*(19)^27 convert it into prime numbers.
(2)^10*(3)^10*(17)^17*(19)^27.
2,3,17 and 19 are all prime numbers. So, their factors is 10+10+17+27 = 64.
Example 10: What is the remainder when 3^997 is divided by 4?
Simple way-> Solution: 997 is divided by 4. It gives 1 as remainder.
So, 3^1, 3^5, 3^9 like this way we should do. So that's give the answer 3,243,19683..
In everything we divide by 4 means it gives 3 as remainder.
So 3 is the answer.
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