Hexadecimal Equations (Problem 6 6 , Version 4 4 )

( ( 2 ( 5 ) + 4 ( 3 ) ) × 4 6 ( 3 ) ) F ( 4 ) 5 A = ((2^{(5)} + 4^{(3)}) \times 46^{(3)})F - (4)_5A = ?

Give your answer in denary.

Note:

x ( 0 ) x^{(0)} denotes the rising factorial

( x ) 0 (x)_0 denotes the falling factorial

For more information about rising and falling factorials, check out the Wikipedia article .

Hint: Perform the calculations in the brackets then remove the brackets.

Bonus: In the discussion, convert the answer to hexadecimal.


The answer is 36263003156.

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1 solution

Yajat Shamji
Jul 13, 2020

2 ( 5 ) = 2 ( 2 + 1 ) ( 2 + 2 ) ( 2 + 3 ) ( 2 + 4 ) = 2 ( 3 ) ( 4 ) ( 5 ) ( 6 ) = 720 2^{(5)} = 2(2 + 1)(2 + 2)(2 + 3)(2 + 4) = 2(3)(4)(5)(6) = 720

4 ( 3 ) = 4 ( 4 + 1 ) ( 4 + 2 ) = 4 ( 5 ) ( 6 ) = 120 4^{(3)} = 4(4 + 1)(4 + 2) = 4(5)(6) = 120

720 + 120 = 840 720 + 120 = 840

4 6 ( 3 ) = 46 ( 46 + 1 ) ( 46 + 2 ) = 46 ( 47 ) ( 48 ) = 103776 46^{(3)} = 46(46 + 1)(46 + 2) = 46(47)(48) = 103776

840 × 103776 = 87171840 840 \times 103776 = 87171840

( 4 ) 5 = 4 ( 4 1 ) ( 4 2 ) ( 4 3 ) ( 4 4 ) = 4 ( 3 ) ( 2 ) ( 1 ) ( 0 ) = 0 (4)_5 = 4(4 - 1)(4 - 2)(4 - 3)(4 - 4) = 4(3)(2)(1)(0) = 0

A = 10 A = 10

F = 15 F = 15

87171840 F 10 87171840F - 10

8 × 1 6 8 + 7 × 1 6 7 + 1 × 1 6 6 + 7 × 1 6 5 + 1 × 1 6 4 + 8 × 1 6 3 + 4 × 1 6 2 + 0 × 1 6 1 = 8 × 4294967296 + 7 × 268435456 + 1 × 16777216 + 7 × 1048576 + 1 × 65536 + 8 × 4096 + 4 × 256 + 0 × 16 = 36263003151 + 15 = 36263003166 8 \times 16^8 + 7 \times 16^7 + 1 \times 16^6 + 7 \times 16^5 + 1 \times 16^4 + 8 \times 16^3 + 4 \times 16^2 + 0 \times 16^1 = 8 \times 4294967296 + 7 \times 268435456 + 1 \times 16777216 + 7 \times 1048576 + 1 \times 65536 + 8 \times 4096 + 4 \times 256 + 0 \times 16 = 36263003151 + 15 = 36263003166

36263003166 10 = 36263003156 36263003166 - 10 = 36263003156

Answer: 36263003156 \rightarrow \fbox{36263003156}

For the Bonus:

36263003156 36263003156

Convert to hexadecimal:

871718414 871718414

Bonus Answer: 871718414 \rightarrow \fbox{871718414}

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