Number System - Study Mode
[#251] If (10 12 + 25) 2 - (10 12 - 25) 2 = 10 n , then the value of n is :
Correct Answer
(C) 14
Explanation
Solution: (10 12 + 25) 2 - (10 12 - 25) 2 = 4 × 10 12 × 25 [∵ (a + b) 2 - (a - b) 2 = 4ab] = 10 12 × 100 = 10 12 × 10 2 = 10 14 Hence, n = 14
[#252] Which of the following numbers is divisible by 3, 7, 9 and 11 ?
Correct Answer
(B) 2079
Explanation
Solution: Clearly, 639 is not divisible by 7 Consider 2079 Sum of its digits = (2 + 0 + 7 + 9) = 18 So, it is divisible by both 3 and 9 Also, (79 - 2) = 77, which is divisible by 7 So, 2079 is divisible by 7 Also, (9 + 0) - (7 + 2) = 0 So, 2079 is divisible by 11 Hence, 2079 is divisible by each one of 3, 7, 9 and 11
[#253] When a certain positive integer P is divided by another positive integer, the remainder is $${r_{1}}$$ . When a second positive integer Q is divided by the same integer, the remainder is $${r_{2}}$$ and when (P + Q) is divided by the same divisor, the remainder is $${r_{3}}$$ . Then the divisor may be :
Correct Answer
(D) $${r_{1}}$$ + $${r_{2}}$$ - $${r_{3}}$$
Explanation
Solution: Let P = x + $${r_{1}}$$ and Q = y + $${r_{2}}$$, where each of x and y are divisible by the common divisor. Then, P + Q = (x + $${r_{1}}$$) + (y + $${r_{2}}$$) = (x + y) + ($${r_{1}}$$ + $${r_{2}}$$) (P + Q) leaves remainder $${r_{3}}$$ when divided by the common divisor. ⇒ [(x + y) + ($${r_{1}}$$ + $${r_{2}}$$) - $${r_{3}}$$] is divisible by the common divisor. Since (x + y) is divisible by the common divisor, so divisor = $${r_{1}}$$ + $${r_{2}}$$ - $${r_{3}}$$
[#254] For the integer n, if n 3 is odd, then which of the following statements are true ? I. n is odd II. n 2 is odd III. n 2 is even
Correct Answer
(C) I and II only
Explanation
Solution: n 3 is odd ⇒ n is odd and n 2 is odd ∴ I and II are true.
[#255] The unit digit in the product 7 71 × 6 63 × 3 65 = ?
Correct Answer
(D) 4
Explanation
Solution: 7 71 × 6 63 × 3 65 ↓ ↓ ↓ Unit Place 7 3 6 3 3 1 ↓ ↓ ↓ Unit Digit ⇒ 3 × 6 × 3 = 54 ⇒ 4 is answer