If m and n are positive integers, what is the greatest common factor of \(7m\) and \(14n\)?
(1) m is an odd integer
(2) \(n = m + 14\)
m > 0 and n > 0 both integers. We need to find the greatest dividing number common to both ‘7m’ and ‘14n’.
Statement 1) \(m = 1, 3, 5, 7\) and so on
We know that if \(m < 7\) then \(GCF(7m, 14n) = 7\)
Or if \(m = 7\) i.e. a multiple of 7 then \(GCF(7m, 14n) = 49\) or more depending on value of ‘n’.
But nothing is given about ‘n’ so INSUFFICIENT.
Statement 2) \(n = m + 14\)
\(n – m = 14\)
which means either n and m can take any value. So,
if both ‘n’ and ‘m’ take values other then multiple of 7 like 15 and 1, 29 and 15 respectively then \(GCF(7m, 14n) = 7\)
but if both ‘n’ and ‘m’ take values as multiple of 7 like 21 and 7, 49 and 35 respectively then \(GCF(7m, 14n) = 49\) or more depending on the values of ‘m’ and ‘n’ both.
As ‘n’ and ‘m’ can take any value so INSUFFICIENT.
Together 1) and 2)
We have m as odd i.e. 1, 3, 5, 7, 9 …. so on and
accordingly n becomes 15, 17, 19, 21, 23 …. So on. Though nothing new is given so INSUFFICIENT.
However, on checking for n and m having values 15 and 1, 17 and 3, 19 and 5 respectively \(GCF(7m, 14n) = 7\)
But if n and m take values 21 and 7 then \(GCF(7m, 14n) = 49\)
Thus INSUFFICIENT.
Answer (E).
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