Number System - Study Mode

[#171] How many composite numbers ae there from 53 to 97?
Correct Answer

(A) 35

Explanation

Solution: 53 to 97 (Composite number) = 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96 Total number = 35

[#172] The sum of the odd divisors of 216 is:
Correct Answer

(C) 40

Explanation

Solution: 216 → 2 3 × 3 3 = (2 0 + 2 1 + 2 2 + 2 3 ) × (3 0 + 3 1 + 3 2 + 3 3 ) Sum of odd factors = 1 + 3 + 9 + 27 = 40

[#173] What least value which should be added to 1812 to make it divisible 7, 11 and 14?
Correct Answer

(B) 36

Explanation

Solution: 7, 11, 14, LCM = 154 154k = 154 × 12 = 1848 1848 - 1812 = 36

[#174] For any integral value of n, 3 2n + 9n + 5 when divided by 3 will leave the remainder
Correct Answer

(B) 2

Explanation

Solution: 3 2n + 9n + 5 Put n = 1 ⇒ 3 2 × 1 + 9 × 1 + 5 ⇒ 9 + 9 + 5 ⇒ 23 ⇒ $$frac{{23}}{3}$$ ⇒ remainder = 2 Note: value of n can be 1, 2, 3, 4, . . . . .

[#175] If a certain number of two digit is divided by the sum of its digits, the quotient is 6 and the remainder is 3. If the digits are reversed and the resulting number is divided by the sum of the digits, the quotient is 4 and the remainder is 9. The sum of the digits of the number is
Correct Answer

(C) 12

Explanation

Solution: Let the number be 10x + y Dividend = Divisor × Quotient + Remainder ∴ 10x + y = 6(x + y) + 3 ⇒ 10x + y = 6x + 6y + 3 ⇒ 10x - 6x + y - 6y = 3 ⇒ 4x - 5y = 3 . . . . . . (i) Again, 10y + x = 4(x + y) + 9 ⇒ 10y + x = 4x + 4y + 9 ⇒ 6y - 3x = 9 ⇒ 2y - x = 3 . . . . . . (ii) ∴ By equation (i) + 4 × (ii), 4x - 5y = 3 8y - 4x = 12 $$overline {3{ ext{y}},,,,,,,,, = 15} $$ ⇒ y = 5 From equation (ii) 2 × 5 - x = 3 ⇒ x = 10 - 3 ⇒ x = 7 ∴ Sum of digits = x + y = 7 + 5 = 12