Question: Solve the equation (16x - {x})[x] = 2024 where [x] and…

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Question: Solve the equation (16x - {x})[x] = 2024 ﻿where [x] ﻿and…

Anu Bhardwaj · Accepted Answer

We are given the equation: ( 16 x − { x } ) [ x ] = 2024 Where: [ x ]  is the greatest integer less than or equal to x (floor function) { x } = x − [ x ] is the fractional part of  Step-by-step solution: Let: [ x ]  =  a , where  a  ∈ Z So,  { x }  =  x − a Therefore: x = a + { x } Now substitute this into the equation: ( 16 x − { x } ) [ x ] = 2024 (16x - \{x\})[x] = 2024 ( 16 ( a + { x } ) − { x } ) ⋅ a = 2024 (16(a + \{x\}) - \{x\}) \cdot a = 2024 ( 16 a + 16 { x } − { x } ) ⋅ a = 2024 (16a + 16\{x\} - \{x\}) \cdot a = 2024 ( 16 a + 15 { x } ) ⋅ a = 2024 (16a + 15\{x\}) \cdot a = 2024 Now distribute: 16 a 2 + 15 a { x } = 2024 16a^2 + 15a\{x\} = 2024 Since a a a is an integer and { x } \{x\}  is a fraction in the range [ 0 , 1 ] [0, 1) , try various integer values of a a a to make the equation work. Try  a = 11 16 ( 11 ) 2 + 15 ( 11 ) { x } = 2024 ⇒ 1936 + 165 { x } = 2024 ⇒ 165 { x } = 88 ⇒ { x } = 88 165 = 8 15 16(11)^2 + 15(11)\{x\} = 2024 \Rightarrow 1936 + 165\{x\} = 2024 \Rightarrow 165\{x\} = 88 \Rightarrow \{x\} = \frac{88}{165} = \frac{8}{15} ​ This is valid since  { x } = 8 15 < 1 \{x\} = \frac{8}{15} < 1 ✅ Final Answer: x = [ x ] + { x } = 11 + 8 15 = 173 15 x = [x] + \{x\} = 11 + \frac{8}{15} = \frac{173}{15} ​ ✔️ So the solution is: x = 173 15 \boxed{x = \frac{173}{15}} ​ ​

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Question: Solve the equation (16x - {x})[x] = 2024 \ufeffwhere [x] \ufeffand {x} \ufeffare the greatest integer less than or equal to x and the fractional part of x respectively.

Anu Bhardwaj

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