Number System - Study Mode

[#356] The difference of the squares of two consecutive even integers is divisible by which of the following integers ?
Correct Answer

(B) 4

Explanation

Solution: Let the two consecutive even integers be 2n and (2n + 2). Then, (2n + 2) 2 - (2n) 2 = (2n + 2 + 2n) (2n + 2 - 2n) = 2(4n + 2) = 4(2n + 1), which is divisible by 4

[#357] Which one of the following is a prime number ?
Correct Answer

(C) 373
(G) 373

Explanation

Solution: 437 > 22 All prime numbers less than 22 are : 2, 3, 5, 7, 11, 13, 17, 19 161 is divisible by 7, and 221 is divisible by 13 373 is not divisible by any of the above prime numbers. Therefore 373 is prime

[#358] The sum of even numbers between 1 and 31 is:
Correct Answer

(C) 240

Explanation

Solution: Let S n = (2 + 4 + 6 + ..... + 30). This is an A.P. in which a = 2, d = 2 and l = 30 Let the number of terms be n Then, a + (n - 1)d = 30 ⇒ 2 + (n - 1) x 2 = 30 ⇒ n = 15 $$eqalign{
& herefore {S_n} = frac{n}{2}left( {a + I}
ight) cr
& ,,,,,,,,,,,,, = frac{{15}}{2} imes left( {2 + 30}
ight) cr
& ,,,,,,,,,,,,, = 15 imes 16 cr
& ,,,,,,,,,,,,, = 240 cr} $$

[#359] If (64) 2 - (36) 2 = 20 × x, then x = ?
Correct Answer

(D) 140

Explanation

Solution: $$eqalign{
& 20 imes x = left( {64 + 36}
ight)left( {64 - 36}
ight) cr
& 20 imes x = 100 imes 28 cr
& herefore {kern 1pt} x = frac{{100 imes 28}}{{20}} cr
& ,,,,,,,,,,, = 140 cr} $$

[#360] What is the unit digit in the product (3 65 × 6 59 × 7 71 )?
Correct Answer

(C) 4

Explanation

Solution: Unit digit in 3 4 = 1 ⇒ Unit digit in (3 4 ) 16 = 1 Therefore Unit digit in 3 65 = Unit digit in [ (3 4 ) 16 × 3 ] = (1 × 3) = 3 Unit digit in 6 59 = 6 Unit digit in 7 4 ⇒ Unit digit in (7 4 ) 17 is 1 Unit digit in 7 71 = Unit digit in [(7 4 ) 17 × 7 3 ] = (1 × 3) = 3 Therefore Required digit = Unit digit in (3 × 6 × 3) = 4