a^37 ≡ a (mod 1729).

Use Euler’s theorem to establish the following:(a) For any int…

                              

Use Euler’s theorem to establish the following:

(a) For any integer a, a^37 ≡ a (mod 1729).
[Hint: 1729 = 7·13·19.]

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(b) For any integer a, a^13 ≡ a (mod 2730).
[Hint: 2730 = 2 · 3 · 5 · 7 · 13.]

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(c) For any odd integer a, a33 ≡ a (mod 4080).
[Hint: 4080 = 15·16·17.]

      Step 1 of 5

a) For any integer a, we need to establish

From Euler’s theorem when .
Now put
So, by Euler’s theorem
By the properties of divisibility,

Put , so, by Euler’s theorem,

Put , so, by Euler’s theorem

Since 7, 13, 19 are pair-wise relatively prime, therefore,

Step 2 of 5

b) We need to show that
From Euler’s theorem when
Now put

Putting while 3 is a prime number.
So, by Euler’s theorem,

Put

Put

Put

Since 2, 3, 5, 7 and 13 are pair wise relatively prime, by the properties of congruences that if ,

• Step 3 of 5
  c) We need to show that for any odd integer a,
  
  From Euler’s theorem when
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• Step 4 of 5
  Now, when
  
  When
  
  
  When
  
  It implies,