The cube of an odd number is odd. So only 50653 is possible. We can also check the answer by calculating the cube of 37
= 37 $\times$ 37  $\times$ 37  = 50653.

By prime factorization, we have

$343=7\times 7\times 71331=11\times 11\times 11512=8\times 8\times 881=3\times 3\times 3\times 3$

For a number to be a perfect cube ,each of its factors must be repeated thrice.
343, 1331 and 512 are the perfect cubes of 7, 11 and 8 respectively, whereas 81 is not, as 3 is repeating 4 times which is not a triplet.

81 is not a perfect cube.

For a number to be a perfect cube, its prime factors should be in the form of triplets
By prime factorization, 43200 =  $2^6$× $3^3$× $5^2$

Only two 5’s are there so we need one more 5 to make a triplet.

So if 43200 is multiplied by 5, we get a perfect cube.

100 = ${10}^2$
= $1^3$+ $2^3$+ $3^3$+ $4^3$
= 1+8+27+64

The average of 3 and 7 is 5. The cube of 5 is 125. The cube of 3 and 7 is 27 and 343 respectively. The average of 27 and 343 is 185 which is not the cube of 5.