Number System

Let number of pigeons be 'X'.
so, number of rabbit's = 90 - X
we know that,
2X + 4(90 - X) = 224
2X + 360 - 4X = 224
X = 68
Hence, there are 68 pigeons in the zoo.

LCM of 7,10,15,21,28 is: 
so, 2 ∗ 2 ∗ 3 ∗ 5 ∗ 7 = 420
⇒ 99999 / 420
This gives remainder as 39
So, 99999 − 39 = 99960

We have
4 cases of first digit {2,4,6,8}
5 cases of second digit {0,2,4,6,8}
But 44 is one case we have to omit. So total ways for leftmost two digit's are 4×5 - 1 = 19
5 cases of third digit {1,3,5,7,9}
5 cases of fourth digit {1,3,5,7,9}
5 cases of fifth digit {1,3,5,7,9}
So total ways = 19×5×5×5 = 2375

45 minutes = $$\frac{3}{4}$$ of an hour.
An hour = 1 / 24 of day $$\frac{3}{4}$$ of hour
= 3/ (4×24) of day
= 1/32 of the day.

Remainder = (73 x 75 x 78 x 57 x 197 x 37) / 34

If 73 is divided by 34, it will give remainder 5, 75 divided 34 gives remainder 7 and so on.)

= (5 x 7 x 10 x 23 x 27 x 3) / 34
= (35 x 30 x 23 x 27) / 34
= (1 x (-4) x (-11) x (-7)) / 34
= (1 x -4 x -11 x -7) / 34
= (28 x -11) / 34 
= (-6 x -11) / 34
= 66 / 34

452 = 2025, If we add 9 with 2016 we get 2025.

Given Exp. = [(a + b)² - 4ab] = (a-b)²,
where a = 476 and b = 424
[(476 + 424)² - 4×476×424]
= (476 - 424)²
= 52²
= 2704

Given Exp. = (1000)9 / 10^24 = (10^3)^9/10^24 = 10^(27-24) = 10^3 = 1000

$$Given Exp. = a^2 + b^2 - 2ab$$ where a = 287 and b = 269
$$= (a - b)^2 = (287 - 269)^2$$
$$= (18^2)$$
= 324