1.find hcf and lcm of 344 and 60 by primefactoration method. 2.find hcf of 870 aned 225 by euclicd's algorithm. 3.find lcm and hcf of 26 and 91 and verify that lcm*hcf=product of two number. 4.find hcf of 732 and 852 using euclid' diviion algorith. 5.find the hcf f 126 and 96 using the euclid' division algorthm. 6.prove that (6+root2) irrational. 7.find the lcm and hcf of 510 and 92 and verify that lcm*hcf=product orf two numbers. 8.prove that root 5 i irrational. 9.by using euclid division algorithm,find the hcf of 875 and 625. 10.using euclid's division algorithm, find hcf of 165 and 395. 11.show that root3-5 s irrational. 12.show that root2 - 5 i irrational. 13.using euclids dvision algorithm,find the hcf of 26 and 422. 14.prove that root2 is irrational. 15.prove that root11 is an irrational. 16.prove that 2*root3-7 i an irrational.
Answers
1) Find hcf and lcm of 344 and 60 by prime factorization method
First we will write the number as the product of prime numbers
344 = 2x2x2x43
60 = 2x2x3x5
Count the maximum number of each factor appears in either quantity. The product of those factors is the Least Common Multiple (LCM)
Lcm = 2*2*2*43*3*5 = 5160
The product of least powers of common prime factors gives HCF
HCF = 2*2 = 4
2) Find HCF of 870 and 225 by euclicd's algorithm
According to euclicd's algorithm we will write the numbers in the form of a=bq+r
Here a = 870 and b = 225
870 = 225 q + r
Now we will divide 870 by 225, where q is the quotient and r is the remainder
870= 225*3+195 since remainder is not equal to zero
Hence for the step 2, a = 225 and b = 195
225 = 1 * 195 + 30
Similarly 195 = 30 * 6 + 15
And 30 = 15 * 2 + 0
Therefore 15 is the hcf of 870 and 225
8) prove that root 5 is irrational
The number is irrational that is it cannot be expressed as a ratio of integars a and b
To prove that this statement is true, let us assume that
Underoot 5
Is rational that it could be written as a fraction a/b in lowest terms that is in which a and b are integers and do not have a common factor other than 1
This would imply that the number is multiple of 5
Because 5 is the prime number it implies that it is a multiple of 5
A= 5c for some integer c
5 b square = a square = 25 c square
Divided by 5 then this means
B square = 5c square
So the number is a multiple of 5 and just as it did for a then means b is a multiple of 5 but a and and b were presumed to lack a common factor other than 1, so this is a contradiction and the fraction a/b for root 5 must fail to exist
By the contradiction it is proved that underoot 5 is an irrational number.