Problems on Numbers Quiz Set 012

0.19191919... is same as $0.\ov 19$. Let y = $0.\ov 19$. Multiply by 100, 100y = 19 + $0.\ov 19$ which is same as 19 + y. So 99y = 19, ⇒ y = $19/99$. So answer = ${19/99}$.

We can see that $4095000 = 2^3 × 3^2 × 5^4 × 7^1 × 13^1$. The number of prime factors is sum of the powers = 11.

If the above number has to be divisible by 9, the sum of the digits, i.e., 3 + 6 + 3 + 1 + 8 + x + 5 + 9 + 4 + 4, should be divisible by 9. So we can see that $x + 43$ should be divisible by 9. By inspection, x = 2.

Let $x/6 = y/12 = z/28$ = k. Then ${x + y + z}/x$ is same as ${6k + 12k + 28k}/{6k}$ which equals $7{2/3}$.

${23/3}$ is same as $7{2/3}$.

Let the number be N. Then it is given that ${13/20}$ x N = 23. ⇒ N = 23 X ${20/13}$. ⇒ ${23/13}$ x N = ${23/13}$ x 23 X ${20/13}$ = ${10580/169}$.

${23/13}$ is same as $1{10/13}$. ${10580/169}$ is same as $62{102/169}$.