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In a certain number system there are only two notations to represent numbers: @ and %. 0 is represented by @ and 1 by %. The subsequent numbers are represented in the following manner: 2 is represented as %@, 3 is represented as %%, 4 is represented as %@@, 5 is represented as %@% and so on. Based on this coded language, answer the following questions: What number is represented by %%@%@%%?

In a certain number system there are only two notations to represent numbers: @ and %. 0 is represented by @ and 1 by %. The subsequent numbers are represented in the following manner: 2 is represented as %@, 3 is represented as %%, 4 is represented as %@@, 5 is represented as %@% and so on. Based on this coded language, answer the following questions: What number is represented by %%@%@%%? Correct Answer 107

According to the given information,

Number

0

1

2

3

4

5

Code

@

%

%@

%%

%@@

%@%

Simplified Code

0

1

10

11

100

101

 

So, this entire code language is based on only 2 symbols @ and %, i.e. 0 and 1.

Let’s work out the logic behind the arrangement of these digits.

On observation, we notice that starting from the rightmost digit, each digit is multiplied with corresponding power of 2, starting with 20, and then all these terms are added to obtain the number in decimal system.

2 → 10 ⇒ (0 × 20) + (1 × 21) = (0 × 1) + (1 × 2) = 0 + 2 = 2

3 → 11 ⇒ (1 × 20) + (1 × 21) = (1 × 1) + (1 × 2) = 1 + 2 = 3

4 → 100 ⇒ (0 × 20) + (0 × 21) + (1 × 22) = (0 × 1) + (0 × 2) + (1 × 4) = 0 + 0 + 4 = 4

5 → 101 ⇒ (1 × 20) + (0 × 21) + (1 × 22) = (1 × 1) + (0 × 2) + (1 × 4) = 1 + 0 + 4 = 5 and so on.

Based on this logic,

%%@%@%% ↔ 1101011

1101011 ⇒ (1 × 20) + (1 × 21) + (0 × 22) + (1 × 23) + (0 × 24) + (1 × 25) + (1 × 26)

= (1 × 1) + (1 × 2) + (0 × 4) + (1 × 8) + (0 × 16) + (1 × 32) + (1 × 64)

= 1 + 2 + 0 + 8 + 32 + 64

= 107

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