Question

If two positive integers a and b are written as a= x5y2 and b=x2y3, x and y are prime numbers then find the hcf (a,b)

Answer 1

since x and y are prime they cannot be factorized further
a=x5y2 and b=x2y3
1st dividing continuously by x
a/x=x4y2⇒a/x2=x3y2⇒....⇒a/x5=y2 cannot be further divided by x
for b
b/x=xy3⇒b/x2=y3  cannot be further divided by x
for a its x5 and for b its x2
HCF=x2
same for y
for a its y2 and for b its y3
HCF=y2
HCF(a,b)=x2*y2
               =x2y2

Answer 2

Concept

The Highest common factor of the HCF, as the name suggests, is the highest number that divides each of the given numbers.

Given

two positive integers a and b such that,

a = x^5y^2 and b = x^2y^3

Find

we need to find the HCF of a and b

Solution

We have,

a = x^5y^2 and b = x^2y^3

dividing a by x^5 throughout we get

a/(x^5) = y^2

dividing b by x^2 throughout we get

b/(x^2) =y^3

since x and y are prime numbers they cannot be simplified further.

Thus the highest x factor of a and b is x^2 only.

Similarly for the y term we have,

a/(y^2) = x^5

and b/(y^3) = x^2

⇒ the highest y factor is y^2

Thus, the HCF of a and b will be x^2y^2

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