Question

Can two number have 16 as HCF and 380 as LCM? Give reasons.

Answer 1

Given two numbers a and b , let p1, p2, … pn be all the primes which divide either a or b .

Now we can write the unique prime factorisations of a and b :

a = p1a1⋅p2a2⋯pnan

b = p1b1⋅p2b2⋯pnbn

where each ai and bi is a non-negative integer.

HCF(a,b) = ∏i=1npimin(ai,bi)

LCM(a,b) = ∏i=1npimax(ai,bi)

We define a set of non-negative integers ci where

ci = max(ai,bi)−min(ai,bi) = |ai−bi|

so that now we can write

LCM(a,b)HCF(a,b) = ∏i=1npici

which is clearly an integer.

Thus the LCM must be divisible by the HCF .

It is not possible to have HCF=16 and LCM=380 because 380 is not divisible by 16 .

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