Number System - Study Mode

[#531] What is the remainder when 4 61 is divided by 51 ?
Correct Answer

(D) None of these

Explanation

Solution: 4 61 = 4 × 4 60 = 4 × (4 4 ) 15 = 4 × (256) 15 Now, (x n - a n ) is divisible by (x - a) for all values of n. ∴ (256 15 - 1) is divisible by (256 - 1) i.e., 255 and hence by 51 ⇒ On dividing (256) 15 by 51, we get 1 as remainder ⇒ On dividing 4 60 by 51, we get 1 as remainder ⇒ On dividing 4 61 by 51, remainder obtained = (4 × 1) = 4

[#532] If n is a natural number and n = $${p_1}^{{x_1}}$$ xa0 $${p_2}^{{x_2}}$$ xa0 $${p_3}^{{x_3}}$$ where p 1 , p 2 , p 3 are distinct prime factors, then the number of prime factors for n is :
Correct Answer

(B) $${x_1} imes {x_2} imes {x_3}$$

Explanation

Solution: Given n = $${p_1}^{{x_1}}{p_2}^{{x_2}}{p_3}^{{x_3}}$$ Where p 1 , p 2 , p 3 are distinct prime factors Number of prime factors form : = (x 1 × x 2 × x 3 ) = x 1 x 2 x 3 Hence option (B) is correct

[#533] The number of prime factors in the expression 6 10 × 7 17 × 11 27 is equal to :
Correct Answer

(B) 64

Explanation

Solution: 6 10 × 7 17 × 11 27 = (2 × 3) 10 × 7 17 × 11 27 = 2 10 × 3 10 × 7 17 × 11 27 Number of prime factors in the given expression = (10 + 10 + 17 + 27) = 64

[#534] A number when divided by 195 leaves a remainder 47. If the same number is divided by 15, the remainder will be :
Correct Answer

(B) 2

Explanation

Solution: Let the number be x and the quotient be q Then, x = 195q + 47 = (15 × 13q) + (15 × 3) + 2 = 15 (13q + 3) + 2 So, the given number when divided by 15 gives 2 as remainder.

[#535] A young girl counted in the following way on the fingers of her left hand. She started calling the thumb 1, the index finger 2, middle finger 3, ring finger 4, little finger 5, then reversed direction, calling the ring finger 6, middle finger 7, index finger 8, thumb 9 and then back to the index finger for 10, middle finger for 11, and so on. She counted upto 1994. She ended on her
Correct Answer

(B) index finger

Explanation

Solution: Number of thumbs = 1, 9, 17, 25, ..... This is an AP in which a = 1 and d = (9 - 1) = 8 ∴ T n = a + (n - 1) d = 1 + (n - 1) 8 = (8n - 7) ∵ 8n - 7 = 1994 ⇒ 8n = 2001 ⇒ n = 250 T 250 = 1 + (250 - 1) 8 = 1 + 249 × 8 = 1993 So, 1993 lies on thumb and 1994 on index finger.