Grade 10 RealNumber (HCF,LCM)

Real Number _ HCF, LCM

Grade : 10

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1. A fruit vendor has 990 apples and 945 oranges. He packs them into baskets. Each basket contains only one of the two fruits but in equal number. Find the number of fruits to be put in each basket in order to have minimum number of baskets.


2. Find HCF of 81 and 237 and express it as a linear combination of 81 and 237
i.e. HCF  (81, 237) = 81x + 237y for some x and y.


3. State Fundamental theorem of Arithmetic. Is it possible that HCF and LCM of two numbers be 24
and 540 respectively. Justify your answer.

4. Given that HCF(306,657)= 9. Find LCM of (306,657) ??

5.A shopkeeper has 120 litres of petrol 180 litres of Diesel And 240 litres of kerosene. He wants to sell oil by filling with three kinds of oil in tin of equal capacity what should be the greatest capacity of such a tin.

6.  144 cartons of Coke cans and 90 cartons of Pepsi cans are to be stacked in a canteen if each stack is of the same height and is to contain carton of the same drink what would be the greater number of cartons each stack would have ?

7. Find the largest number which divides 245 and 1029 leaving remainder 5 in each case ?

8. The HCF of two numbers is 23 and their LCM is 1449. If one of the numbers is 161 , find the other?

9.Find HCF and LCM of 378, 180 and 420 by prime factorisation method. Is HCF × LCM of these numbers equal to the product of the given three numbers?

10.Find the HCF and LCM of 510 and 92 and verify that HCF × LCM = Product of two given numbers.

Answers

1. HCF of 990 and 945 is 45.

2.  x =- 38 and y = 13. These values of x and y are not unique.

3. Fundamental theorem of Arithmetic :  states that every positive integer (except the number1) can be represented in exactly one way apart from rearrangement as a product of one or more primes .

LCM of two numbers should be exactly divisible by their HCF. In other words LCM is always a multiple of HCF. Since, 24 does not divide 540 two numbers cannot have their HCF as 24 and LCM as 540.

HCF = 24

LCM = 540

HCF / LCM  = 24 / 540 = = 22.5 not an integer.

4. 22338

5. HCF= 60 litres

6. HCF =18

7. HCF = 16

8. 207

9. HCF × LCM  not equal to Product of three numbers.

10. proof. HCF×LCM = Product of two numbers