Statement 1 alone does not answer the question, since some integers ending in 9 are perfect squares (9, 49) and some are not (29, 39).

Statement 2 alone does not answer the question, since of the numbers in that range, only 169 is a perfect square (of 13).

From both statements together, however, it follows that the number in question is 169, which, as just stated, is the square of 13.

Statement 1 is a giveaway; the only prime numbers between 50 and 60 are 53 and 59.

Statement 2 is not a giveaway, though, since there are at least two sums of primes equal to 112:

 and 

From Statement 1 it follows that  is a multiple of 5, but it does not answer the question by itself; 5 is prime, but the other multiples of 5 are composite. Similarly, from Statement 2 it follows that  is a multiple of 7, but, 7 is prime and the other multiples of 7 are composite.

If we can assume both statements, then, since 5 and 7 are relatively prime, it follows that  is a multiple of both 5 and 7; since  now has at least four factors (1,5, 7, and 35),  is composite.

Statement 1 is not enough to answer the question. For example, the integers can be 3, 4, and 5, the product of which is 60, which is divisible by 4; or, the integers can be 5, 6, and 7, the product of which is 210, which is not divisible by 4. 

From Statement 2, however, it can be inferred that the product is divisible by 4. If two of the integers are even, the two can be written as  and  for some integers . If we call the third integer , the product is , and the product must be divisible by 4.

Any group of three consecutive integers must include at least one multiple of 3 and at least one even number (multiple of 2). Therefore, if Statement 1 is assumed, 2 and 3 both divide the product of the integers and, subsequently, so does 6. 

Statement 2 is not enough, however - for example, the product of 3, 4, and 5 is 60, a multiple of 3, but the product of 4, 5, and 7 is 140, not a multiple of 3.

The only criterion for a number to be divisible by 4 is that the last two digits must form a number divisible by 4. This makes Statement 1 unhelpful. Statement 2, however, proves the number to not be divisible by 4, since 58 is not a multiple of 4.

For a number to be divisible by 11, the alternating sum of its digits must be divisible by 11. We do not have enough information from either statement to tell us one way or the other. 

Assume both statements are true. Let  be the digit that replaces the circle and the triangle; let  be the digit that replaces the square and the diamond. Then the alternating sum is

,

which is divisible by 11. Therefore, the resulting number is also divisible by 11.

Let  be the missing digit.

If we assume only Statement 1, then the absolute value alternating sum of the digits must be a multiple of 11. That is:

  is a multiple of 11. The only positive value of  that does this is , which makes .

If we assume only Statement 2, then the sum of the digits must be a multiple of 3. That is,

 is a multiple of 3. This happens if , so Statement 2 only narrows the last digit down to three possibilities.

We have 3 pieces of rope, and their lengths all add to 2 ft, or 24 inches.  Let X, Y, and Z represent the three lengths.  Our equation is:

Let's take the first condition.  Let's say .  Then, substituting into our equation, we get:

, or

.

That is not sufficient to give us unique values for y and z.

The second condition, when transformed into math, tells us:

.

We can substitute this into the original equation, 

,

and get:

Which means that .

Then, we know that ,

or, 

.

But this also does not give us unique values for x and y.

But if we use BOTH of the statements, we know that z= 10 inches from the second statement, and x= 8 from the first statement, so we can solve for the last remaining rope length, and find that y=6.

Both of the statements together are necessary to solve the problem.

Using Statement (1):

In order to determine that 10-n is even, we need to know whether n is even or odd. If n is even, we can write:

In that case, m-n will be even. If n is odd, m-n will be odd as well.

Therefore, Statement (1) does not provide sufficient information to determine whether m-n is even.

Using Statement (2):

Although m>n, m and n could be even or odd integers. If m and n are both even, m-n will be even. If m and n are both odd, then m-n will be even. However, if either m or n is an even integer and the other is an odd integer, then m-n will be odd.

Statement (2) also does not provide enough information to determine whether m-n is even.

Therefore, Both statements TOGETHER are not SUFFICIENT.