The only one even prime number is 2 and this is not in the form of 3k + 1.
Thus any prime number of the form 3k + 1 is an odd number.

Correct Option: B

The only one even prime number is 2 and this is not in the form of 3k + 1.
Thus any prime number of the form 3k + 1 is an odd number.
This means 3k must be even number which implies that k must be even number also.
Let k = 2m.
Then a prime of the form 3k + 1 is of the form 3(2m) + 1 = 6m+1.
Every prime number of form 3k + 1 can be represented in the form 6m + 1 only, when k is even.

Let the two digit number be 10x + y.
Now, according to the question,
x + y = 10............(i)
if digits reversed , number get decreased by 36,
and ( x +10y) = ( 10x + y ) - 36
Solve the equation.

Correct Option: B

Let the two digit number be 10x + y.
Now, according to the question,
x + y = 10............(i)
if digits reversed , number get decreased by 36,
and ( x +10y) = ( 10x + y ) - 36
⇒ - 9y + 9x = 36
⇒ x - y = 4 ..................... (ii)
On adding Eqs. (i) and (ii), we get
2x = 14
⇒ x = 7
∴ x = 7 and y = 3
∴ Required number is 73.
So,the number is a multiple of a prime number.

Let the fraction be x / y.
According to the question,
( x+2)/(y+1) = 1/2
⇒ 2x + 4 = y+1
⇒ 2x - y = -3 .........................(i)
Again according to the question,
(x+1)/(y-2) = 3/5
⇒ 5x +5 = 3y - 6
∴ 5x - 3y = -11..........................(ii)

Correct Option: A

Let the fraction be x / y.
According to the question,
( x+2)/(y+1) = 1/2
⇒ 2x + 4 = y+1
⇒ 2x - y = -3 .........................(i)
Again according to the question,
(x+1)/(y-2) = 3/5
⇒ 5x +5 = 3y - 6
∴ 5x - 3y = -11..........................(ii)
on multiplying Eq. (i) by 3 and then subtracting from Eq. (ii) we get,
multiplying Eq. (i) by 3
6x - 3y = -9
subtracting from Eq. (ii)
5x- 3y - ( 6x - 3y ) = -11 - (-9)
5x- 3y - 6x + 3y = -11 + 9
- x = - 2
x= 2
On putting the value of x in Eq. (i) we get
2x- y = -3
2 x 2 - y = -3
⇒ 2 x 2 -y = -3
⇒ -y = -3 - 4
∴ y = 7
∴ Original fraction = x/y =2/7

Method 1 to solve this question.
Let p be the first odd number.
Then next odd number is p + 2.
According to question,
Sum of five consecutive odd numbers 175
⇒ p + p + 2 + p + 4 + p + 6 + p + 8 = 175

Solve the above equation to get the answer.

Method 2 to solve this question.
Let p be the even number.
Then next odd number is p + 1. so on p + 3 , p + 5................ P + 9
According to question,
Sum of five consecutive odd numbers 175
⇒ p + 1 + p + 3 + p + 5 + p + 7 + p + 9 = 175

Solve the above equation to get the answer.

Correct Option: E

Method 1 to solve this question.
Let p be the first odd number.
Then next odd number is p + 2.
According to question,
Sum of five consecutive odd numbers 175
⇒ p + p + 2 + p + 4 + p + 6 + p + 8 = 175
⇒ 5p + 20 = 175
⇒ p = (175 - 20)/5 = 31
According to question,
Sum of the second largest and square of smallest one = (31 + 6) + (31)²
Sum of the second largest and square of smallest one = 37 + 961 = 998

Method 2 to solve this question.
Let p be the even number.
Then next odd number is p + 1. so on p + 3 , p + 5................ P + 9
According to question,
Sum of five consecutive odd numbers 175
⇒ p + 1 + p + 3 + p + 5 + p + 7 + p + 9 = 175
⇒ 5p + 25 = 175
⇒ p = (175 - 25)/5
⇒ p = (150)/5
⇒ p = 30
First odd number = p + 1 = 30 + 1 = 31
According to question,
Sum of the second largest and square of smallest one = (30 + 7) + (30 + 1)²
Sum of the second largest and square of smallest one = 37 + 961 = 998

Let the third number = p
Then, the first number = 3p
and second number = 3p/2
According to question,
( 3p + 3p/2 + p )/3 = 154

Correct Option: C

Let the third numbers = p
Then, the first number = 3p
and second number = 3p/2
According to question,
( p + 3p + 3p/2 )/3 = 154
⇒ ( 2p + 6p + 3p )/6 = 154
∴ p = 154 x 6/11 = 84
∴ Required difference = 3p - p = 2p
= 2 x 84 = 168