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LCM of 254, 508 = 2x 2x 127 = 508, , Thus, 508 is the smallest common number that is divisible by 254 and 508., Now, as we need remainder 4 while dividing, the required number is 4 more than, the LCM and so, the required number is 508 + 4 =512., , Example 10: what is the smallest 5 digit number that is exactly divisible by 72 and 108?, , , , Solution: — First let us find the LCM of 72 and 108 2 [72108, , (by division method). 2 136,54, 2 18,27, , LCM of 72 and 108= 2x2x2x3x3x3=216 3 low, , Now, all multiples of 216 will also be common multiples of “3 3,9, , 72 and 108. a lis, , The smallest 5 digit number = 10,000. 14, , Now, 10,000 + 216 gives quotient as 46 and remainder as, , 164., , Hence the next multiple of 216 i.e., 216 x 47 = 10,152 is the required smallest 5 digit, number that is exactly divisible by 72 and 108., , | Na, , | ‘6th_Maths Term 2_English Full Book.indb 18 e 09-08-20, , | EEE * ., www.tntextbooks.in, , , , Example 11: There are four Mobile Phones in a house. At 5 a.m, all the four Mobile, Phones will ring together. Thereafter, the first one rings every 15 minutes, the second one, tings every 20 minutes, the third one rings every 25 minutes and the fourth one rings, every 30 minutes. At what time, will the four Mobile Phones ring together again?, , Solution: This is a LCM related sum. So, we need to find the LCM of 15, 20, 25 and 30., , 2 |15, 20, 25, 30, "2_|15, 10, 25, 15, 5 (15, 5, 25,15, 3 (3,1, 5,3, 5/41 5,1, ~ [a4 14, , , , , , The LCM of 15, 20, 25 and 30is2x2x3x5x5, = 300 minutes = 5 x 60 minutes = 5x 1 hour = 5 hours, Thus, the four Mobile Phones will ring together again at 10.00 a.m., , o/|\s, WV aia mts, , A small boy went to a town to sell a basket of wood apples. On the way, some robbers, grabbed the fruits from him and ate them. The small boy went to the King and complained., & The King asked him, “How many wood apples did you bring?”. The boy replied,, “Your Majesty! I didn’t know, but I knew that if you divided my fruits into groups of, 2, one fruit would be left in the basket”. He continued saying that if the fruits were, divided into groups of 3, 4, 5 and 6, the fruits left in the basket would be 2, 3, 4 and 5, respectively. Also, if the fruits were divided into groups of 7, no fruit would be there in, the basket. Can you find the number of fruits, the small boy had initially?, (This problem is taken from the famous Mathematics problems collection book in Tamil, called “Kanakkathikaram” under the heading of “Wood Apple Problem”), , 1.8 Relationship between the Numbers and their HCF and LCM, Let us find the HCF and the LCM of 36 and 48. First, find the factors of 36 and 48 using, division method., , , , , , , , , , , , , , , , 36=2x2x3x3; 48=2x2x2x2x3 2. 36 2 48, , HCF =2x2x3=12 2 18 2 24, , LCM =2x2x3x2x2x3 =144 3 9 2 12., Observe that, 36 x 48 = 144 x 12 = 1728 3 (3 2 |6, , e@ We find that, ‘a 3/3, product of two given numbers = their HCF x LCM 1, , In general, for any 2 numbers x and y,, xx y = HCF (x, y) x LCM (x, y)