## PLAYING WITH NUMBERS - ESSENTIAL POINTS

• Factors of a number are numbers what we can multiply to get the number OR with which it is exactly divisible. For example 2 × 3 = 6, so 2 and 3 are factors of 6. The factors of the number 90 are 1, 3, 9, 18. If we divide the number 90 by these numbers, then it is divisible exactly.
• Factors are usually positive or negative whole numbers (no fractions), so ½ × 24 = 12 is not factor.
• The factors of a number are limited.
• The factor of every number is 1 and the number itself. For instance the number 6 is divisible by 1 and 6. You can check it with any number.
• Multiples are what we get after multiplying the number by an integer (not a fraction).
• For example the multiples of the number 7 are 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84. These all numbers come in table of 7.
• The multiples of a number are unlimited.
• The prime number is the number which is divisible by itself, and the number 1 only. 5 is a prime number, as it is divisible by only 1 and 5.
• The composite numbers are the numbers which have factors other than 1 and the number itself. 6 is a composite number because besides 1 and 6, it has the factors 2 and 3.
• The even numbers are the numbers which are divisible by 2. For instance 2, 4, 6, 8, 10, 12, 20, 100.
• The odd numbers are the numbers which are not divisible by 2. For instance 5, 9, 11, 13, 21, 27, 105.
• All the even numbers are composite numbers except 2. 2 has factors only 1 and the number 2 itself. 4 has factor, 1,2 and 4.
• The smallest prime number is 2. Hence its reciprocal is 1/2.
• The Least Common Multiple (LCM) of two numbers is the smallest number which is a multiple of each of the numbers. For instance if there are two numbers 12 and 30. The least common multiple of these numbers is 60, as 60 is the smallest number which is a multiple of both 12 and 30. The number 120 is also a common multiple of 12 and 30, but it is not lowest.
• The Highest Common Factor (HCF) of two or more numbers is the greatest or the largest among the common factors. For example, HCF os numbers 24 and 36 is 12. There are other common factors like 4, 6 etc, but 12 is the highest.
• Two numbers are said to be co–prime if their greatest common factor is 1. For instance 13 and 15 are co prime, becaue 1 is the common factor of both of them.
• The Highest Common Factor of two numbers is product of two numbers/ Least Common Multiple of two numbers. Let the two numbers be 40 and 60. Their L.C.M is 120. Hence H.C.F is 40 X 60/ 120 = 20.
• Divisibility Rules
• Divisibility Rule for 2 - If it's unit's digit is 0,2,4,6 or 8. For example, 36 is divisible but not 39.
• Divisibility Rule for 3 - If the sum of it's digits is divisible by 3. For example, 165 (1+ 6 + 5 = 12 ÷3 = 4) is divisible by 3.
• Divisibility Rule for 4 - If the number formed by its digit in ten's and unit's place is divisble by 4. For example, 636 (36 ÷ 4 = 9) is divisible by 4.
• Divisibility Rule for 5 - If its unit's digit is 0 or 5. For example, 250 , 375 is divisible by 5 but not 208.
• Divisibility Rule for 6 - If it is divisible by both 2 and 3. For example, 114 (it is even, and 1+1+4=6 and 6÷3 = 2) hence it is divisible by 6.
• Divisibility Rule for 7 - If you double the last digit and subtract it from the rest of the number and the answer is either 0 or divisible by 7. For example 672 (Double 2 is 4, 67-4=63, and 63÷7=9), hence it is divisible by 7.
• Divisibility Rule for 8 - A number is divisible by 8, if the number formed by its digits in hundred's,ten's and units places are divisible by 8. For example 109816 (816÷8=102), hence it is divisible by 8.
• Divisibility Rule for 9 - A number is divisible by 9,if the sum of its digits is divisible by 9. For example 1629 (1+6+2+9=18, and again, 1+8=9), hence it is divisible by 9.
• Divisibility Rule for 10 - A number is divisible by 10 if its unit's place digits is zero. For example 240, 350, 470 are divisible by 10 but 245, 366, 475 are not divisible.
• Divisibility Rule for 11 - A number is divisible by 11, if the difference of the sum of its digits in odd places and the sum of its digits in even places is eiether 0 or multiple of 11. For example, 1364 ((3+4) - (1+6) = 0),  3729 ((7+9) - (3+2) = 11) are both divisible by 11.