If $$\sqrt[3]{7^a\times 35^{b+1} \times 20^{c+2}}$$ is a whole number then which one of the statements below is consistent with it?

In the given statement, the expression becomes a whole number only when the powers of all the prime numbers are also whole numbers. 

Let us first simplify the expression a bit by expressing all terms in terms of prime numbers. 

 $$\sqrt[3]{7^a\times 35^{b+1} \times 20^{c+2}}$$

$$\Rightarrow \sqrt[3]{7^a\times 5^{b+1} \times 7^{b+1} \times 2^{2(c+2)} \times 5^{c+2}}$$

$$\Rightarrow \sqrt[3]{2^{2c+4} \times 5^{b+c+3} \times 7^{a+b+1}}$$

$$\Rightarrow 2^{\frac{2c+4}{3}} 5^{\frac{b+c+3}{3}} 7^{\frac{a+b+1}{3}} $$

Now, from the given options, we can put in values of the variables and check the exponents of all the numbers. 

Option A : a = 2, b = 1, c = 1 : 

In this case, we can see that exponent of 5 ie $$\frac{b+c+3}{3} = \frac{5}{3} $$ is not a whole number. 

Option B : a = 1, b = 2, c = 2

In this case, we can see that exponent of 2 ie $$\frac{2c+4}{3} = \frac{8}{3} $$ is not a whole number. 

Option C : a = 2, b = 1, c = 2

In this case, we can see that exponent of 2 ie $$\frac{2c+4}{3} = \frac{8}{3} $$ is not a whole number. 

Option D : a = 3, b = 1, c = 1

In this case, we can see that exponent of 5 ie $$\frac{b+c+3}{3} = \frac{5}{3} $$ is not a whole number. 

Option E : a = 3, b = 2, c = 1

In this case, we can see that all exponents are whole numbers. 

Thus, option E is the correct option.