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Highest Common Factors Highest common factor is the greatest number which will divide exactly into two or more numbers. For example 4 is the highest common factor (HCF) of 20 & 24. i.e. 20 1, 2, (4), 5, 10, 20 24 1, 2, 3, (4), 6, 8, 12, 24 Example 1: Find the H.C.F of 24 & 78 Method 1 Express each number as a product of its prime factors Workings 2 24 2 78 2 12 2 36 2 6 2 18 3 3 3 9 3 3 2423x3 78(23 x 3) x 3 The H.C.F. is the product of the common prime factors. HCF23x3 8x324 Method II 242x2x2x3 782x2x2x3x3 Common factor2x2x2x3 HCF24 LCM: Lowest Common Multiple Multiples of 2 are 2,4,6,8,10,12,14,16,18,20,22,24 Multiples of 5 are 5,10,15,20,25,30,35,40 Notice that 10 is the lowest number which is a multiple of 2 & 5.10 is the lowest common multiple of 2& 5 Find the LCM of 20, 32, and 40 Method 1 Express each number as a product of its prime factors 2022x5 3225 4022x2x5 The prime factors of 20, 32 and 40 are 2 & 5 .The highest power of each prime factor must be in the LCM These are25 and 5 Thus LCM 25 x5 160 Method II 2 20 32 40 2 10 16 20 2 5 8 10 4 5 4 5 5 5 1 5 1 1 1 LCM 2 x 2 x 2 x 4 x 5 160 Class work Find the HCF of: (1) 28 and 42 (2) 504 and 588 (3) Find the LCM of 84 & 210 PERFECT SQUARES A perfect square is a whole number whose square root is also a whole number .It is always possible to express a perfect square in factors with even indices. 9 3x3 25 5x5 225 15x15 3x5x3x5 32 x 52 9216 96 2 32 x 32 2 32 x 42 X 82 32x24 x26 32 x2 10 Workings 2 9216 2 4608 2 2304 2 1152 2 576 2 288 2 144 2 72 2 36 2 18 3 9 3 3 9216 32x210 Example Find the smallest number by which the following must be multiplied so that their products are perfect square 540 252 Solution 2 540 2 270 3 135 3 45 3 15 5422 x 33x 5 5 5 1 The index of 2 even. The index of 3 and 5 are odd .One more 3 and one more 5 will make all the indices even. The product will then be a perfect square .The number required is 3x5 15 2 252 2 126 3 63 3 21 7 7 1 252 22x32x7 Index of 7 is odd, one more 7 will make it even. Indices i.e. 22x 32x 72 Therefore 7 is the smallest numbers required WEEKEND ASSIGNMENT The lowest common multiple of 4, 6 and 8 is (a) 24 (b) 48 (c) 12 (d) 40 Find the smallest number by which 72 must be multiplied so that its products will give a perfect square (a) 3 (b) 2 (c) 1 (d) 5 The lowest common multiple of 4, 6 and 8 is (a) 24 (b) 48 (c) 12 (d) 40 The H.C.F. of 8, 24 and 36 is ___ (a) 6 (b) 4 (c) 18 (d) 20 The L.C.M. of 12, 16 and 24 is ___ (a) 96 (b) 48 (c) 108 (d) 24 THEORY Find the smallest number by which 162 must be multiplied so that its product will give a perfect square. Find the HCF and L.C.M. of the following figures 30 & 42 64 & 210 See also WHOLE NUMBERS AND DECIMAL NUMBERS BASIC OPERATION OF INTEGERS WEIGHT VOLUME OF CYLINDER AREA OF RIGHT ANGLED TRIANGLE

H.C.F & L.C.M AND PERFECT SQUARES