Problems on Numbers Quiz Set 001

0.75757575... is same as $0.\ov 75$. Let y = $0.\ov 75$. Multiply by 100, 100y = 75 + $0.\ov 75$ which is same as 75 + y. So 99y = 75, ⇒ y = $75/99$. So answer = ${25/33}$.

Let the number be ab. When it is reversed and added to itself we get (10a + b) + (10b + a) = 11 × (a + b). We are given 143 = 11 × (a + b) ⇒ $a + b = 143 / 11 = 13$, so the digits are $a$ and $13 - a$. We are given their product as a × (13 - a) = 36, which is a quadratic expression that can be simplified to $(a - 4) × (9 - a) = 0$. So the number could be 49 or 94.

Let the numbers be n - 1, n and n + 1. The sum is 3n. We are given 3n = 696, ⇒ n = $696/3$, i.e., n = 232. So the smallest is 231.

The first term is 150, common difference is d = -10, n-th term is 0. So $0 = 150 + (n - 1) × -10$ which gives $n = 1 + {150/10} = 16$.

We can see that $16200 = 2^3 × 3^4 × 5^2$. The number of prime factors is sum of the powers = 9.