If m and n are positive integers, what is the greatest common factor

If m and n are positive integers, what is the greatest common factor of 7m and 14n?

(1) m is an odd integer
we don't know what n is
insufficient

(2) n = m + 14
GCF of 7m and 14(m+14)
m could be any number
the GCF could be m and m could be any number
insufficient

together
odd integer m could still be any number

therefore, E

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).

gcf(7m,14n)=7•gcf(m,2n)

(1) m is an odd integer: different answers, insufic.
if n = m, then gcf(m,2m)=m
if m=3 and n=6, then gcf(3,2•6)=3
if m=5 and n=6, then gcf(5,2•6)=1

(2) n = m + 14: different answers, insufic.
if m=3 and n=17, then gcf(m,2n)=(3,34)=1
if m=4 and n=18, then gcf(m,2n)=(4,36)=4

(1&2) n = m + 14, and m = odd: different answers, insufic.
if m=3 and n=17, then gcf(m,2n)=(3,34)=1
if m=7 and n=21, then gcf(m,2n)=(7,42)=7

Answer (E)

IMO its E
(1) m is an odd integer
(2) n = m + 14

Because even after combining we will get multiple values as a common factor.
Case 1) m= 3 , n = 17; then 7 is the GCF
Case 2) m=7, n=21; then 49 is the GCF

We are to find the GCF of 7m and 14n where m and n are positive integers.

Statement 1 is insufficient because it only says that m is an odd integer. m can take many values and there is no further information about n which we only know is a positive integer.

Statement 2 is also insufficient because knowing that n=m+14 is not enough to get unique values of 7m and 14n
We get 7m and 14(m+14).
When m=1, we get 7 and 210. Their GCF =7
When m=2, we get 14 and 224 and their GCF=14. Hence not sufficient.

Both statements taken together is also not sufficient. Because we get 7m and 14(14+m).
We know m is an odd number which leads to many possibilities such as 1,3,5,7,9,...etc. Each possible value of m leads to a different GCF. Hence not sufficient.

The answer is therefore E imo.

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