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Question
A number a has 7 as its smallest prime factor, and another number b has 3 as its smallest prime factor. What is the smallest prime factor of the number a−b?
A number a has 7 as its smallest prime factor, and another number b has 3 as its smallest prime factor. What is the smallest prime factor of the number a−b?
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Solution
The correct option is B 2
Since 7 is the smallest prime factor of a, it is clear that 2 is not a factor of a. Thus, a is an odd number, i.e. a=2n+1 for any n.
Similarly, when 3 is the smallest prime factor of b, 2 cannot be a factor of b and thus b is also odd.i.e., b=2m+1 for any m.
Hence we have a−b=(2n+1)−(2m+1)=2(n−m)=2k ( taking 2 common and assuming k=n−m.)
We can thus see that 2 is a factor of a−b, and hence 2 is the smallest prime factor of a−b.
2
Since 7 is the smallest prime factor of a, it is clear that 2 is not a factor of a. Thus, a is an odd number, i.e. a=2n+1 for any n.
Similarly, when 3 is the smallest prime factor of b, 2 cannot be a factor of b and thus b is also odd.i.e., b=2m+1 for any m.
Hence we have a−b=(2n+1)−(2m+1)=2(n−m)=2k ( taking 2 common and assuming k=n−m.)
We can thus see that 2 is a factor of a−b, and hence 2 is the smallest prime factor of a−b.
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