CHAPTER 3 PLAYING WITH NUMBERS
FACTORS When a number is divided by its factor, the remainder is zero. A factor of a number completely divides the number. MULTIPLES A number is a multiple of its factors. E.g. 6x12=12. Therefore 12 is a multiple of both 2 and 6. Some inferences that can be drawn are:
PERFECT NUMBER A number whose sum of factors is twice the number is called a perfect number. E.g. factors of 6= 1,2,3,6. And 1+2+3+6=12. 12 is twice the number 6. Therefore 6 is a perfect number. PRIME AND COMPOSITE NUMBERS A prime number is a number which has exactly two factors 1 and the number itself. E.g. 2, 3, 5, 7, 9, etc. A composite number is a number which has more than two factors. E.g. 4, 6, 9, 8, etc. Note: 1 is neither a prime nor a composite number. TEST FOR DIVISIBILITY OF NUMBERS
COMMON FACTORS Consider two numbers 6 and 15. Factors of 6= 1, 2, 3, 6 Factors of 15= 1, 3, 5, 15 1 and 3 are the common factors of 6 and 15. COPRIME NUMBERS If two numbers have only 1 as a common factor, they are called coprime numbers. E.g. Consider 5 and 16. Factors of 5= 1, 5 Factors of 16= 1, 2, 4, 8, 16 Common factor= 1 Therefore, 5 and 16 are coprime numbers. COMMON MULTIPLES If a number is a multiple of two or more numbers, it is said to be their common multiple. SOME DIVISIBILITY RULES
PRIME FACTORISATION When a number is expressed as a product of its prime factors, it is called prime factorisation. E.g. 12= 2x2x3 and 28= 2x2x7 HIGHEST COMMON FACTOR (HCF) The HCF of two or more numbers is the highest of their common factors of the number. It is also called Greatest Common Divisor (GCD). HCF can be calculated using prime factorization. E.g. HCF of 20, 24, 36 20= 2x2x5 24= 2x2x2x3 36= 2x2x3x3 We take the common factor i.e. 2 and it occurs twice therefore, HCF= 2x2= 4 LOWEST COMMON MULTIPLE (LCM) The lowest among the common multiple of two or more numbers is called their LCM. Finding the LCM of 16 and 20 using prime factorization: 16= 2x2x2x2 20= 2x2x5 LCM is the product of the prime factors counted the number of times they occur. Here, LCM= 2x2x2x2x5= 80
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