In mathematics, the greatest common divisor (gcd) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4.
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Trick 109 – Find HCF Mentally without Factorization
A common multiple is one that is a multiple of 2 or more than 2 numbers. For example:
136 = 23 x17 144 = 24 x 32 168= 23x3x7 Distinct factors are 2, 17, 3 and 7. The highest power of 2 is 4, of 3 is 2, of 17 is 1 and of 7 is 1. So, LCM = 24 x 32 x17 x7 = 17136 2. Method 2:
Illustrations: Example: There are some boxes lying in a straight line. Every 6th box contains a muffin, every 8th contains a chocolate and every 9th contains a soft-toy. Which is the first box to have all 3 items? a) 48 b) 36 c) 432 d) none of these Solution Muffins will be contained in boxes- 6, 12, 18, 24, 36…. Chocolates will be contained in boxes- 8,16,24,32…. Soft-toys will be contained in boxes- 9,18,27,36…. The first box which contains all three items will have to be a multiple of 6, 8 and 9. Being the first box to contain all three items, it will be the lowest multiple of all 3, which is the LCM. LCM of 6, 8, 9 is 72. So, this is the answer.
Greatest Common divisor (GCD), also called HCF, is the largest integer that perfectly divides two or more given numbers. For Example: The HCF of 18 and 12 is 6. 6 is the largest number that divides both 18 and 12. How To Find Out HCF: Method 1: Prime Factorization Method-
Example: To find the HCF of 136,144, 168 To find the HCF of 136,144, 168 Step 1: 136 = 23 x17 144 = 24 x 32 168= 23x3x7 Step 2: Distinct factors present = 2,3,7,17 Step 3: Raising each factor to the minimum present (i.e. 23,30,70 and 170) HCF= 23=8 Method 2: Division Method
Example: To find the HCF of 12 and 15 To find the HCF of 12 and 15 15/12|R = 3 12/3|R=0 Thus, the HCF= 3 HCF and LCM of Fractions HCF of fractions = HCF of numerators/ LCM of denominators LCM of fractions = LCM of numerators/ HCF of denominators LCM x HCF = Product of two numbers (this can be applied only for 2 numbers) Question: The circumference of the wheels of a vintage car are 7/3 and 13/4 m respectively.A mark is made on each of these wheels at their point of contact with the ground.Find the distance traveled by the car before which the part of the wheels with the marks is again on the ground at the same time next time. a) 85 m b) 183 m c) 91m d) None of these Solution: Option (c) LCM of 7/3 and 13/4 gives the answer = LCM of numerators/ HCF of denominators = 91/1= 91.
Solution: Find the LCM of 4,6,8,10 and 12 = 120 A multiple of 120 less than 2000= 1920 thus, this is the number which is divisible by 4,6,8,10 and 12 Find the LCM of 4,6,8,10 and 12 = 120 A multiple of 120 less than 2000= 1920 thus, this is the number which is divisible by 4,6,8,10 and 12.
Solution: 45=5×32 50=52×2 30=2x3x5 LCM of 45,50 and 30 =52x2x32 = 450 Largest 4 digit number = 9999 9999/450 leaves a remainder of 99 Thus, 9999-99=9900
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