Let the middle integer of the three be \(\displaystyle x\). The three integers are therefore 

\(\displaystyle x-2, x, x+2\), and they can be found using the equation

\(\displaystyle \left (x-2 \right )+ x +\left ( x+2 \right ) = 924\)

\(\displaystyle 3x= 924\)

\(\displaystyle x = 308\)

The three even integers are therefore 306, 308, and 310, and the product of the least and greatest of these is 

\(\displaystyle 306 \times 310= 94,860\)

Let the middle integer of the three be \(\displaystyle x\). The three integers are therefore 

\(\displaystyle x-2, x, x+2\), and they can be found using the equation

\(\displaystyle \left (x-2 \right )+ x +\left ( x+2 \right ) = 537\)

\(\displaystyle 3x= 537\)

\(\displaystyle x = 1 79\)

The three integers are 177, 179, and 181, and the product of the least and greatest is 

\(\displaystyle 177 \times 181 = 32,037\)

Call the least of the four integers \(\displaystyle x\). The four integers are therefore

\(\displaystyle x, x+1, x+2, x+3\),

and they can be found using the equation

\(\displaystyle x + \left (x+1 \right )+\left (x+2 \right )+\left ( x+3 \right )= 3,350\)

\(\displaystyle 4x+6= 3,350\)

\(\displaystyle 4x=3,344\)

\(\displaystyle x=3,344 \div 4 = 836\)

The integers are 836, 837, 838, 839. 

To get the correct response, multiply: 

\(\displaystyle 837 \times 838 = 701,406\)

Let the middle integer of the three be \(\displaystyle x\). The three integers are therefore 

\(\displaystyle x-1, x, x+1\), and they can be found using the equation

\(\displaystyle \left (x-1 \right )+ x +\left ( x+1 \right ) = 427\)

\(\displaystyle 3x= 427\)

\(\displaystyle x = 142\frac{1}{3}\)

This contradicts the condition that the numbers are integers. Therefore, three integers satisfying the given conditions cannot exist. 

Let the middle integer of the three be \(\displaystyle x\). The three integers are therefore 

\(\displaystyle x-1, x, x+1\), and they can be found using the equation

\(\displaystyle \left (x-1 \right )+ x +\left ( x+1 \right ) = 312\)

\(\displaystyle 3x= 312\)

\(\displaystyle x = 104\)

The integers are therefore 103, 104, 105. The correct response is their product, which is

\(\displaystyle 103 \times 104 \times 105= 1,124,760\)

This is a geometric sequence since the pattern of the sequence is through multiplication.  

You have to multiple each value by \(\displaystyle 3\) to get the next one.  

The value before \(\displaystyle x\) is \(\displaystyle 9\) so \(\displaystyle 9*3=27\).

Let \(\displaystyle N\), \(\displaystyle N+1\), \(\displaystyle N+2\), and \(\displaystyle N+3\) be the four consecutive integers. Then their sum would be 

\(\displaystyle N + (N+1) + (N+2) + (N+3) = 4N + 6\)

In other words, if 6 were to be subtracted from their sum, the difference would be a multiple of 4. Therefore, we subtract 6 from each of the choices and see if any of the resulting differences are multiples of 4.

\(\displaystyle 178 -6 = 172 ; 172 \div4 = 43\)

\(\displaystyle 213 -6 = 207 ; 207 \div4 = 51 \; R \; 3\)

\(\displaystyle 420 -6 = 414 ; 414 \div4 = 103 \: R \: 2\)

\(\displaystyle 143 -6 = 137 ; 137 \div4 = 34\: R\: 1\)

Since this only happens in the case of 178, this is the only number of the four that can be a sum of four consecutive integers: 43, 44, 45, 46.