Question

prove that 5^6n-3^6n is divisible by 152 for n€N​

Answer 1

To prove --->

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( 5 ⁶ⁿ - 3 ⁶ⁿ) is divisible by 152

Proof --->

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Let P( n ) = ( 5⁶ⁿ - 3⁶ⁿ ) is divisible by 152

P(1 ) = 5⁶ˣ¹ - 3⁶ˣ¹

= 5⁶ - 3⁶

= 15625 - 729

= 14896

= 152 × 98

So P(n ) is divisible by 152 for n= 1

Now let P(n ) is divisible by 152 for n=k

P(k ) = 5⁶ᵏ - 3⁶ᵏ

Now we have to prove that P( n ) is divisible by 152 for n=k+1

P(k+1) =5⁶(ᵏ⁺¹) - 3⁶(ᵏ⁺¹)

= 5⁶ᵏ⁺⁶ - 3⁶ᵏ⁺⁶

= 5⁶ᵏ 5⁶ - 3⁶ᵏ 3⁶

Adding and subtracting 5⁶ 3⁶ᵏ

=5⁶ 5⁶ᵏ - 5⁶ 3⁶ᵏ +5⁶ 3⁶ᵏ - 3⁶ᵏ 3⁶

= 5⁶ ( 5⁶ᵏ - 3⁶ᵏ ) + 3⁶ᵏ ( 5⁶ - 3⁶)

= 5⁶ P ( k ) +3⁶ᵏ ( 15625 -729)

= 5⁶ P( k ) + 3⁶ᵏ ( 14896)

= 5⁶ P( k ) + 3⁶ᵏ ( 152 × 98 )

So P(k+1) is divisible by 152 if P(k) is divisible by 152

So by principle of mathematical induction P(n) is divisible by 152 for any natural number

Answer 2

Answer:

Step-by-step explanation:

5^6n - 3^6n = (5^3n + 3^3n) (5^3n - 3^3n)

When n = 1

Then it becomes (5³ +3³) (5^3n - 3^3n)  

Hence it will be divisible by 152.