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Higher Level of Difficulty in LCM and HCF

In our last post Shortcuts to find HCF and LCM, we learn how to solve simple HCF and LCM in less time, however, it was just an initial step. In exams, you may or may not ask easy problems on the concepts. High level of Difficulty always persists in Fraction and Remainder topic, needs full proficiency and of course focus on the level of speed.



HCF and LCM



HCF and LCM of decimal fractions:


Problem 1: Find HCF and LCM of 0.63, 1.05 and 2.1

Solution:


Step1: Make the same number of decimals like 0.63, 1.05 and 2.10
Step2: Now find the HCF of 63, 105 and 210 leaving decimals. HCF comes out 21.
Step3: So, HCF of 0.63, 1.05 and 2.10 is 0.21
Step4: Find the LCM of 63,105 and 210 leaving decimals again. LCM comes out 630
Step5: So, the LCM of 0.63, 1.05 and 2.10 is 6.30

Comparison of Fractions:


When Fractions are to be compared finding greater, lesser, equal or not equal. Method to follow:
  1. First, find the LCM of the denominator.
  2. Multiply the numerator and denominator of the numbers with the same number to make the denominator equal to LCM.
  3. The resulting fraction with the greatest numerator is the greatest.
Problem 1: Arrange the fractions 17/18, 31/36, 43/45, 59/60 in ascending order.

Solution:


Step1: Find the LCM of the denominator. LCM of 18,36,45,60=180
Step2: Now Multiply the numerator and denominator of the numbers with the same number to make the denominator equal to LCM.

17/18= (17x10) / (10x18) =170/180
31/36= (31x5) / (36x5) = 155/180
43/45= (43x4) / (45x4) = 172/180
59/60 = (59x3) / (60x3) = 177/180

Step3: Denominators are the same, so arrange numerators in the sequence asked.
Step4: 155 < 170 < 172
<177 155="" 170="" 172="" 177="" br="" so=""> Step5: Hence 31/36 < 17/18 < 43/45 < 59/60 in ascending order


Remainder based problems:


Problem1: The largest number which divides 25,73 and 97 to leave the same remainder in each case.

Solution:


Step1: Subtract the lower number from higher as it gives no harm to HCF

HCF of x, y is G then HCF of x, x-y or x+y, x-y is also G

Step2: It becomes

73-25 =48, 97-73=24, 97-25=72

Step3: The largest number is given in question so we find the HCF of 48,24,72.
Step4: HCF=24 is the largest number that divides all numbers to leave the same remainder

Problem2: What is the least number which divided by 8,9,12 and 15 leaves the same remainder 1 in each case.

Solution:


Step1: Least number is given means to find LCM
Step2: LCM of 8,9,12 and 15 = 360
Step3: 360+1=361 is the least number which, when divided by 8, 9, 12 and 15 leaves the remainder 1 in each case.

Problem3: The smallest number which, when increased by 5 is divided by 24, 32, 36, 54 is

Solution:


Step1: Least number is given means to find LCM
Step2: LCM of 24, 32, 36, 54 = 867
Step3: In problem, the least number 867 is increased by 5. To get the actual number, subtract 5 from the number

867-5=859

Step4: 859 is the smallest number which, when increased by 5 is divided by 24, 32, 36, 54

Problem4: The smallest number which, when diminished by 3 is divided by 21, 28, 36, 45 is

Solution:


Step1: Least number is given means to find LCM
Step2: LCM of 21, 28, 36, 45 = 1260
Step3: In problem, the least number of 1260 is diminished by 3. To get the actual number to add 3 to the number

1260 + 3=1263

Step4: 1263 is the smallest number which, when diminished by 3 is divided by 21, 28, 36, 45

Problem5: The smallest number which, when divided by 18, 27 and 36 leaves the remainder 5, 14, 23 is

Solution:


Step1: Subtract remainders from 18,27 and 36 to get the common remainder

18-5=13, 27-14=13, 36-23=13

Step2: Least number asked means to find LCM
Step3: LCM of 18,27,36 =108
Step4: Subtract the common remainder from 108 -13 = 95
Step5: 95 is the smallest number which, when divided by 18, 27 and 36 leaves the remainder 5, 14, 23


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