Number System - Study Mode

[#576] The number (2 48 - 1) is exactly divisible by two numbers between 60 and 70. The numbers are :
Correct Answer

(A) 63 and 65

Explanation

Solution: (2 48 - 1) = [(2 6 ) 8 - 1] = (64) 8 - 1 When n is even , (x n - a n ) is completely divisible by both (x - a) and (x + a) ∴ (64 8 - 1 8 ) is divisible by both (64 - 1) and (64 + 1) ⇒ (2 48 - 1) is divisible by both 63 and 65

[#577] When the square of any odd number, greater than 1, is divided by 8, it always leaves remainder :
Correct Answer

(A) 1

Explanation

Solution: Let the number be N = 2x + 1 N 2 = (2x + 1) 2 = 4x 2 + 1 + 4x = 4x (x + 1) + 1 Clearly, 4x (x + 1) is always divisible by 8 since one of x and (x + 1) is even which when multiplied by 4 is always divisible by 8. Hence, required remainder = 1

[#578] Find the least 6-digit number which is exactly divisible by 349 ?
Correct Answer

(A) 100163

Explanation

Solution: The least 6-digit number = 100000 On dividing 100000 by 349, we get 186 as remainder. Required number : = 100000 - (349 - 186) = 100000 + 163 = 100163

[#579] The value of 5 2 + 6 2 + .... + 10 2 + 20 2 is :
Correct Answer

(A) 755

Explanation

Solution: = 5 2 + 6 2 + .... + 10 2 + 20 2 = (1 2 + 2 2 + 3 2 +..... + 10 2 ) - (1 2 + 2 2 + 3 2 + 4 2 ) + 400 = $$frac{1}{6}$$ n(n + 1) (2n + 1) - (1 + 4 + 9 + 16) + 400, where n = 10 = $$left( {frac{1}{6} imes 10 imes 11 imes 21}
ight)$$ xa0 xa0 - 30 + 400 = (385 - 30 + 400) = 755

[#580] A number when divided by 3 leaves a remainder 1. When the quotient is divided by 2, it leaves a remainder 1. What will be the remainder when numbers is divided by 6 ?
Correct Answer

(C) 4

Explanation

Solution: y = 2 × 1 + 1 = 3 x = 3 × y + 1 = 3 × 3 + 1 = 10 Clearly, 10 when divided by 6, leaves a remainder 4