Given that;
2 < m < p; where
- 'm' and 'p' are integers
- 'm' is not a factor of p
- when 'p' is divided by 'm', there is a remainder called 'r'
Question: is r > 1?
Statement 1: The greater common factor of m and p is 2.
If the HCF of m and p is 2, definitely, 'm' and 'n' are even number.
Since both m and p are greater than 2, and p divided by m will have a remainder that is greater than or equal to 2.
If m=4 and p=6, their HCF = 2
$$and\ \frac{p}{m}=\frac{6}{4}=1\ \frac{2}{4}; \ \ r>1$$
If m=4 and p=10, their HCF = 2
$$and\ \frac{p}{m}=\frac{10}{4}=2\ \frac{2}{4}; \ \ r>1$$
If m=8 and p=14, their HCF = 2
$$and\ \frac{p}{m}=\frac{14}{8}=1\ \frac{6}{8}; \ \ r>1$$
Definitely, r>1; sin ce the target question question can be answered with certainty, statement 1 is SUFFICIENT.
Statement 2: The least common multiple of m and p is 30.
If m=5 and p=6, their LCM = 30
$$and\ \frac{p}{m}=\frac{6}{5}=1\ \frac{1}{5};\ \ \ \ \ r=1$$
If m=10 and p=15, their LCM = 30
$$and\ \frac{p}{m}=\frac{15}{10}=1\ \frac{5}{10};\ \ \ \ \ r>1$$
As seen above, the target question cannot be answered with certainty; hence, statement 2 is NOT SUFFICIENT.
In conclusion, only statement 1 is SUFFICIENT, therefore, option A is the correct answer.
