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.

If we only know that  is divisible by 7, then  may or may not be a multiple of 7. 

Example 1: If , then , which is not divisible by 7.

Example 2: If , then , which is divisible by 7.

A similar argument shows that if we only know that   is divisible by 7, then  may or may not be a multiple of 7. 

Suppose we know both. 

If we know that  is divisible by 7, then, when  is divided by 7, the remainder is 1, and we can write for some integer :

If we know that  is divisible by 7, then, when  is divided by 7, the remainder is 6, and we can write for some integer :

Then 

  for some integer quotient 

So, when  is divided by 7, the remainder is 6, and  is not a multiple of 7. This makes the two statements together sufficient information.

The product of two or more numbers is 0 if and only if at least one of the numbers is 0.

Each of the individual statements eliminates only two possible answers to the question; Statement 1 eliminates  and , and Statement 2 eliminates  and . The two together, however, eliminate all possible answers except for , leaving that to be the number worth zero.

Statement 1 is not sufficient, since there are three primes - 41, 43 and 47 - between 40 and 50.

Statement 2 is not sufficient, since there are numerous pairs of primes that differ by 2 - 3 and 5 are one pair, as are 5 and 7.

Both statements together are sufficient, however; from Statement 1, we know that the primes are chosen from 41, 43, and 47, and from Statement 2, we know that we can only choose 41 and 43.

If the base 5 representation of  ends in 2, then the division of the number by 5 results in a remainder of 2. Since every multiple of 5 ends in 0 or 5, this means that  ends in 2 or 7.

If the base 2 representation of  ends in 1, then the division of the number by 2 results in a remainder of 1. This makes  an odd number, which must end in 1, 3, 5, 7, or 9.

Neither statement alone provides a definitive answer to the question, but both facts taken together do - only a number ending in 7 satisfies both conditions.

There are only two prime numbers between 30 and 40 - they are 31 and 37. Statement 1 provides enough information.

There is only one way to factor 1,147 as the product of two primes:  . Statement 2 is also a giveaway.

A necessary and sufficient condition for a whole number to be divisible by 9 is that the sum of its digits must be divisible by 9. Statement 1 provides proof that this condition is not met, since 26 is not a multiple of 9. Statement 2 provides irrelevant information.