If (a,4)=(b,4)=2, then show that 4 divides a+b.
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Step-by-step explanation:
HCF (a,4)=2 means that 2 is a divisor in a and that 4 isn't. The same goes for b, so a can be written as 4k+2 and b as 4l+2, so a+b=4k+2+4l+2=4(k+l)+4, which clearly has 4 as a divisor.
From the first property, we can write a=2a′, where a′ is odd. From the second property, we can write b=2b′, where b′ is odd. We now have a+b=2(a′+b′). Since a′,b′ are both odd, their sum is even, and 4|(a+b).
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