XAT 2020

Using Fermat's theorem :

If p is a prime number and a, p are co primes $$\left(a^{p-1}\right)\ mod\ p=1$$

Remainder when $$19^{20}$$ is divided by 7 = $$19^2$$ mod 7 =4. ( Here $$19^{20\ }=\ \left(\left(19\right)^6\right)^3\cdot\left(19\right)^2$$

Since the remainder for $$19^6$$ is 1 the remainder for $$\ 19^{20}$$ is  equivalent to the $$\frac{19^2}{7}$$ = 4.

Remainder when $$20^{19}$$ is divided by 7 = $$20^1$$ mod 7 =6.( Here $$\frac{20^6}{7}$$ the remainder is 1 and since 

$$20^{19}=\ \left(20^6\right)^3\cdot\left(20\right)^1\ =\ \frac{\left(1\cdot20\right)}{7}$$. The remainder is 6.

Remainder when $$19^{20} - 20^{19}$$ is divided by 7=4-6=-2=> 5. 

Let's five small drums have a capacity of 1 unit capacity each and one bigger drum has that of 2 units capacity. We need to consider two cases here. One with minimum volume and the other with maximum volume.

Case 1: Minimum value is possible if the bigger drum is half filled. So, total volume of water = 1 + 2 * (1/2) + 2 * (2/3) + 1 = 26/6 ~ 4.3

Case 2: Maximum value is possible if the bigger drum is completely full. So, total volume of water = 2 + 3 * (1/2) + 2 * (2/3) = 29/6 = 4.833

In any case volume of water is more than 4 units and less than 5 units. Hence, exactly 5 smaller drums are adequate to store the water.

FRom ii, China is F.

FRom iv and v, India must be A, Brazil must be G and Russia must be B.

India's GDP due to agriculture = 17% of 2 trillion =0.34 trillion.

FRom i, USA GDP due to agriculture> 0.34 trillion and <1 trillion i.e. it's percentage must be >2% and less than 5.8%

So, USA can be C/E/H.

From point 3, UK must be D and France must be H=> U.S.A must be C/E.

Now, Germany and Japan can be C or I.

6 nations can be uniquely identified. 

FRom ii, China is F.

FRom iv and v, India must be A, Brazil must be G and Russia must be B.

India's GDP due to agriculture = 17% of 2 trillion =0.34 trillion.

FRom i, USA GDP due to agriculture> 0.34 trillion and <1 trillion i.e. it's percentage must be >2% and less than 5.8%

So, USA can be C/E/H.

From point 3, UK must be D and France must be H=> U.S.A must be C/E.

Now, Germany and Japan can be C or I or E.

.'. Everything except option D can be ruled out.

FRom ii, China is F.

FRom iv and v, India must be A, Brazil must be G and Russia must be B.

India's GDP due to agriculture = 17% of 2 trillion =0.34 trillion.

FRom i, USA GDP due to agriculture> 0.34 trillion and <1 trillion i.e. it's percentage must be >2% and less than 5.8%

So, USA can be C/E/H.

From point 3, UK must be D and France must be H=> U.S.A  must be C/E.

Now, Germany and Japan can be C or I.

Option A: If Germany is C, then its GDP will be 1 but given its gdp is 1.2 So, C is not Germany. If Germany is I, then its GDP will be 1.2trillion. Hence, Germany is I => Japan is C/E and USA is C/E. If USA is C then its GDP due to agriculture is not 3rd highest. So, USA is E and Japan is C. 

Option A is correct.