Number System - Study Mode

[#91] Which is not a prime number ?
Correct Answer

(C) 21

Explanation

Solution: 21 = 3 × 7 is not a prime number because 21 is a composite number.

[#92] If x and y are positive integers such that (3x + 7y) is a multiple of 11, then which of the followings is also a multiple of 11 ?
Correct Answer

(A) 5x - 3y

Explanation

Solution: Let 3x + 7y = 11k Then, y = $$frac{(11k - 3x)}{7}$$ Then, $$eqalign{
& = 5x - 3y cr
& = 5x - frac{{3(11k - 3x)}}{7} cr
& = frac{{35x - 33k + 9x}}{7} cr
& = frac{{44x - 33k}}{7} cr
& = frac{{11(4x - 3k)}}{7} cr} $$ Which is divisible by 11

[#93] If the symbol [x] denotes the greatest integer less than or equal to x, then the value of : $$left[ {frac{1}{4}}
ight]$$ $$ + $$ $$left[ {frac{1}{4} + frac{1}{{50}}}
ight]$$ xa0 $$ + $$ $$left[ {frac{1}{4} + frac{2}{{50}}}
ight]$$ xa0 $$ + $$ $$....$$ $$ + $$ $$left[ {frac{1}{4} + frac{{49}}{{50}}}
ight]$$
Correct Answer

(C) 12

Explanation

Solution: Clearly, each of the 38 terms $$left[ {frac{1}{4}}
ight], left[ {frac{1}{4} + frac{1}{{50}}}
ight], left[ {frac{1}{4} + frac{2}{{50}}}
ight], $$xa0 xa0 xa0 $$ ..... $$ $$ , left[ {frac{1}{4} + frac{{37}}{{50}}}
ight]$$ xa0 has a value lying between 0 and 1, While each one of the 12 terms $$left( {frac{1}{4} + frac{{38}}{{50}}}
ight),$$ xa0 $$left( {frac{1}{4} + frac{{39}}{{50}}}
ight),$$ xa0 $$.....,$$ $$left( {frac{1}{4} + frac{{49}}{{50}}}
ight)$$ xa0 has a value lying between 1 and 2. Hence, the given expression = (0 × 38) + (1 × 12) = 12

[#94] If (12 n + 1) is divisible by 13, then n is :
Correct Answer

(C) Any odd integer

Explanation

Solution: (x n + a n ) is divisible by (x + a) when n is odd. ∴ (12 n + 1) is divisible by (12 + 1) i.e., 13 when n is odd.

[#95] By how many of the following numbers is 2 12 - 1 divisible ? 2, 3, 5, 7, 10, 11, 13, 14
Correct Answer

(A) 4

Explanation

Solution: (2 12 - 1) = (4096 - 1) = 4095, when is clearly divisible by 3, 5, 7 and 13 i.e., four numbers in all.