Question

The LCM of two numbers is 45 and their HCF is 5. how many such pairs of numbers are possible?​

Answer 1

Answer:

3 pairs

step-by-step explanation:

As both the numbers have the highest common factor 12 it clearly means that the both the numbers are the multiples if 12,

so let the numbers be 12x and 12y

as given their sum is 84

ATQ,12x+12y=84

12(x+y)=84

x+y=7

so the number of pairs with 7 as their sum are (1,6).(2,5),(3,4)

that is three pairs

the numbers are  (12,72),(24,60),(36,48).

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Answer 2

Answer:

We should know that,

Product of the 2 numbers = LCM × HCF

(This is mathematically proven)

So, LCM = 45 and HCF = 5

Now, let the numbers be 5a and 5b where a and b are coprimes

(Coprimes means there are no other common factors between a and b)

Now, I said 5a and 5b because, now a and b are coprimes and if we take the HCF of a and b we get

HCF = 1

But we need HCF = 5

so, the numbers are 5a and 5b

so, 5a × 5b = 45 × 5

25ab = 225

ab = 225/25 = 9

ab = 9

Now, the coprimes pairs which satisfy ab = 9 is (1, 9)

where a = 1

b = 9

See we can't take (3,3) because they have a common factor of 3, we said a and b are coprimes

So, the possible number is 5(1) and 5(9) = 5 and 45

So, the numbers are 5 and 45

And one and only one pair is possible

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