Number System - Study Mode

[#116] If n is any positive integer, 3 4n - 4 3n is always divisible by :
Correct Answer

(C) 17

Explanation

Solution: Putting n = 1, we get (3 4n - 4 3n ) = (3 4 - 4 3 ) = (81 - 64) = 17 Which is divisible by 17

[#117] When 100 25 - 25 is written in decimal notation, the sum of its digits is :
Correct Answer

(A) 444

Explanation

Solution: 100 25 - 25 = (10 2 ) 25 - 25 = 10 50 - 25 $$ = underbrace {,1000,.....00,}_{50,{ ext{zeros}}} - 25$$ xa0 xa0 $$ = underbrace {,9999,.....,9975,}_{48,{ ext{times}}}$$ ∴ Sum of digits = (48 × 9) + 7 + 5 = 432 + 7 + 5 = 444

[#118] Which is the greatest 5-digit number exactly divisible by 279 ?
Correct Answer

(C) 99882

Explanation

Solution: The greatest 5-digit number = 99999 On dividing 99999 by 279, we get 117 as remainder ∴ Required number = (99999 - 117) = 99882

[#119] If p,q,r are all real numbers then (p - q) 3 + (q - r) 3 + (r - p) 3 is equal to :
Correct Answer

(B) 3(p - q) (q - r) (r - p)

Explanation

Solution: Let a = p - q, b = q - r, c = r - p ∴ a + b + c = p - q + q - r + r - p ⇒ a + b + c = 0 ∴ a 3 + b 3 + c 3 = 3abc ⇒ a 3 + b 3 + c 3 = 3(p - q) (q - r) (r - p)

[#120] If the six digit number 15x1y2 is divisible by 44, then (x + y) is equal to:
Correct Answer

(B) 7

Explanation

Solution: Divisibility law of 4: A number divisible by 4 if its last 2 digit is divisible by 4 Divisibility law of 11: If the difference of the alternating sum of digits of the number is a multiple of 11 (e.g. 2343 is divisible by 11 because 2 - 3 + 4 - 3 = 0, which is a multiple of 11). ⇒ 15x1y2 is divisible by 4 if y = 1 or 3 If y = 1, then ⇒ 15x112 is divisible by 11 if x = 6 If y = 3 then ⇒ 15x132 is divisible by 11 if x = 4 If y = 1 and x = 6, then ⇒ x + y = 1 + 6 = 7 If y = 3 and x = 4, then ⇒ x + y = 3 + 4 = 7