QUERY 51

n is a whole number which when divided by 4 gives 3 as remainder. What will be the remainder when 2n is divided by 4 ?

A) 3
B) 2
C) 1
D) 0

MAHA GUPTA
DIVISION ALGORITHM = DIVIDEND × (DIVISOR × QUOTIENT) + REMAINDER

So, CASE-I: n = 4q + 3.

In case-II the dividend becomes 2n

Thus multiplying the equation of case-I by 2
2n = 8q + 6
=> 2n = 4(2q + 1) + 2

Therefore when 2n is is divided by 4 the remainder is 2 (option ‘B’)

QUERY 52

Which of the following numbers will completely divide (4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴)?

A) 3 
B) 10
C) 11 
D) 13

MAHA GUPTA
4⁶¹ + 4⁶² + 4⁶³ + 4⁶⁴
= 4⁶¹(1 + 4 + 4² + 4³)
= 4⁶¹ × 85

It’s clear from the above that 85 is a factor of the given expression. We know that if a factor of an expression is divisible by any number, that whole expression too is divisible by that number.

But here we see none of the number given in the answer options is dividing 85 completely, so we need to move further like this:

4⁶¹ × 85 = 4⁶⁰ × 4¹ × 85 = 4⁶⁰ × 340

Now, we see 340 is divisible by 10, so the given expression too is divisible by 10 (option ‘B’)