Olympiad Test: Playing With Numbers - 1

5, 10, 26, 50, 122, …
(2)² + 1, (3)² + 1, (5)² + 1,
(7)² + 1, (11)² + 1, (13)² + 1
∴ Next number = (13)² + 1 = 170

63 = 7 × 9
∴ If a number is divisible by 63, then it should be divisible by both 7 and 9. 

Sum of first ‘n’ natural numbers = 
∴ Sum of first 15 natural numbers

    = 15 × 8 = 120

Let ‘x’ be the carry, resulting from sum of b and 3.
8 + 2  + x = 11 ⇒ x = 1
∴ 3 + b = 12
⇒ b = 9

Sum of ‘n’ odd natural numbers (first)= n²
∴ Sum of first 20 odd natural numbers = (20)²
= 400

182 ×  22 = 4004
∴ a = 4

7¹ = 7
7² = 49
7³ = 343
7⁴ = 2401
7⁵ = 16807

∴ After leaving 43 exponents 7ⁿ repeats its unit digit.
∴ 7⁴ⁿ has 1 as its units place digit.
∴ 7³³² has 1 as units place digit.
∴ 7³³³ has 7 as its units place digit. 

1221, 10·12-21 = 99, 99 can be divided by 11, hence 1221 is divisible by 11.

When 8 and 7 are added, 1 is carry.
∴ x = 7 + 1 + 1 = 9
The carry of sum (5 + y) should be 1.
∴ 5 + y = 13
⇒ y = 8
∴ x + y = 17

⇒ a² + a² = 8
⇒ 2a² = 8
⇒ a = 2, b = 4
∴ a × b = 8