Question

Fill up the missing digit. 

If (220)_a = (90)_b then (451)_a = ( )₁₀

Answer 1

It contains 2 & 9, so a and b 2, b 8. The values of a can be 8 or 19. The values of b can be 10 or 16, L.H.S > R.H.S. a 

The possible values of a and b are given below

a	b	Reason
8	10 or 16	a<b and a ≠ b
10	16 only

Case I : 

Let a = 8 and b = 10

ie.(220)₈  ≠ (90)₁₀

So a  ≠ 8 or b  ≠ 10

Case II: 

Let a = 8 and b = 16

(220)₈ = 2x8² +2x8¹ + 0x8⁰

=128+16 = (144)₁₀ 

(90)16 = 9x16¹ + 0x16⁰ = (144)₁₀

So a = 8 and b = 16

Then (451)₈ = 4x8² + 5x8¹ + 1x8⁰

=4x64 + 5x8 + 1

256 + 40 + 1 = (297)₁₀