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.]
(b) For any integer a, a^13 ≡ a (mod 2730).
[Hint: 2730 = 2 · 3 · 5 · 7 · 13.]
(c) For any odd integer a, a33 ≡ a (mod 4080).
[Hint: 4080 = 15·16·17.]
[Hint: 1729 = 7·13·19.]
(b) For any integer a, a^13 ≡ a (mod 2730).
[Hint: 2730 = 2 · 3 · 5 · 7 · 13.]
(c) For any odd integer a, a33 ≡ a (mod 4080).
[Hint: 4080 = 15·16·17.]
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,


From Euler’s theorem


Now put

So, by Euler’s theorem

By the properties of divisibility,


Put



Put



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

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
,


From Euler’s theorem


Now put


Putting

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 5c) We need to show that for any odd integer a,
From Euler’s theoremwhen
Comment(0) - Step 4 of 5Now, when
When
When
It implies,