Question

If a,b,c are integer powers of 2, prove that equation
a^3+ b^4 = c^5
does not have solution, otherwise give a counterexample.

Answer 1

Given : a,b,c are integer powers of 2 ,  

To Find :  prove that equation a^3+b^4=c^5 , does not have solution, otherwise give a counterexample

Solution:

Giving counter example :

(2⁸)³ + (2⁶)⁴  =  (2⁵)⁵

Let say

a = 2^p

b= 2^q

c = 2^r

a³ + b⁴ =  

 a³  = (2^p)³  = 2^(3p)

b⁴ = (2^q)⁴  = 2^(4q)

3p = 4q  is the condition to have only 2 as prime  factor

Hence 12k   will satisfy this

2^12k + 2^12k = 2^(12k+1) = 2^5r

12k+ 1 = 5r

=>  k = 2    r = 5  is the first solution    further can be k = 7  , r = 17 and so on.

p = 8   q  = 6

(2⁸)³  + (2⁶)⁴    = 2²⁴ + 2²⁴   = 2²⁴(1 + 1)  = 2²⁵  = (2⁵)⁵

Hence a³ + b⁴ = c⁵

if a = 2⁸   , b = 2⁶  , c = 2⁵

Learn More:

a^3+b^4=c^5 ,

https://brainly.in/question/39048154

Answer 2

Answer:

Given : a,b,c are integer powers of 2 ,  

To Find :  prove that equation a^3+b^4=c^5 , does not have solution, otherwise give a counterexample

Solution:

Giving counter example :

(2⁸)³ + (2⁶)⁴  =  (2⁵)⁵

Let say

a = 2^p

b= 2^q

c = 2^r

a³ + b⁴ =  

 a³  = (2^p)³  = 2^(3p)

b⁴ = (2^q)⁴  = 2^(4q)

3p = 4q  is the condition to have only 2 as prime  factor

Hence 12k   will satisfy this

2^12k + 2^12k = 2^(12k+1) = 2^5r

12k+ 1 = 5r

=>  k = 2    r = 5  is the first solution    further can be k = 7  , r = 17 and so on.

p = 8   q  = 6

(2⁸)³  + (2⁶)⁴    = 2²⁴ + 2²⁴   = 2²⁴(1 + 1)  = 2²⁵  = (2⁵)⁵

Hence a³ + b⁴ = c⁵

if a = 2⁸   , b = 2⁶  , c = 2⁵

Learn More:

a^3+b^4=c^5 ,

https://brainly.in/question/39048154

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