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Number Theory Level 3

Given a quarternary number 21320 2 4 213202_4 , convert the number into a hexadecimal base.

Submit your answer in a hexadecimal form.

Note: In general, you can't submit an answer with the letters on Brilliant. But this problem is somewhat special that its answer includes letters too. For an example, if your answer is 25 E 25E , then submit it as it is.


The answer is 9E+2.

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Manuel Kahayon by Manuel Kahayon , 21, Philippines

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1 solution

Manuel Kahayon
Jul 8, 2016

213202 213202 in base 4 4 is equal to ( 2 ⋅ 4 5 ) + ( 1 ⋅ 4 4 ) + ( 3 ⋅ 4 3 ) + ( 2 ⋅ 4 2 ) + ( 0 ⋅ 4 ) + ( 2 ⋅ 4 0 ) (2 \cdot 4^5) + (1 \cdot 4^4)+ (3 \cdot 4^3) + (2 \cdot 4^2) + (0 \cdot 4) + (2 \cdot 4^0)

Grouping the terms into pairs, we get that it is also equal to 1 6 2 ( 2 ⋅ 4 + 2 ) + 16 ( 3 ⋅ 4 + 1 ) + 2 16^2 ( 2 \cdot 4 +2) + 16 (3 \cdot 4 +1) + 2

= 1 6 2 ( 9 ) + 16 ( 14 ) + 2 = 9 E 2 16 = 16^2 ( 9) + 16 (14) + 2 = 9E2_{16}

So, our answer is 9 E 2 \boxed{9E2}

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