Sunday, March 31, 2019

a^37 ≡ a (mod 1729).

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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^37a (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, a33a (mod 4080).
[Hint: 4080 = 15·16·17.]

      Step 1 of 5
a) For any integer a, we need to establish
6348-7.3-1P-i1
From Euler’s theorem 6348-7.3-1P-i2 when 6348-7.3-1P-i3 .
Now put 6348-7.3-1P-i4
So, by Euler’s theorem 6348-7.3-1P-i5
By the properties of divisibility, 6348-7.3-1P-i6
6348-7.3-1P-i7
Put 6348-7.3-1P-i8 , so, by Euler’s theorem, 6348-7.3-1P-i9
6348-7.3-1P-i10
Put 6348-7.3-1P-i11 , so, by Euler’s theorem 6348-7.3-1P-i12
6348-7.3-1P-i13
Since 7, 13, 19 are pair-wise relatively prime, therefore,
6348-7.3-1P-i14

Step 2 of 5
b) We need to show that 6348-7.3-1P-i15
From Euler’s theorem 6348-7.3-1P-i16 when 6348-7.3-1P-i17
Now put 6348-7.3-1P-i18
6348-7.3-1P-i19
Putting 6348-7.3-1P-i20 while 3 is a prime number.
So, by Euler’s theorem,
6348-7.3-1P-i21
Put 6348-7.3-1P-i22
6348-7.3-1P-i23
Put 6348-7.3-1P-i24
6348-7.3-1P-i25
Put 6348-7.3-1P-i26
6348-7.3-1P-i27
Since 2, 3, 5, 7 and 13 are pair wise relatively prime, by the properties of congruences that if 6348-7.3-1P-i28 ,
6348-7.3-1P-i29



  • Step 3 of 5
    c) We need to show that for any odd integer a,
    6348-7.3-1P-i30
    From Euler’s theorem 6348-7.3-1P-i31 when 6348-7.3-1P-i32
    • Step 4 of 5
      Now, when 6348-7.3-1P-i33 6348-7.3-1P-i34
      6348-7.3-1P-i35
      When 6348-7.3-1P-i36
      6348-7.3-1P-i37
      6348-7.3-1P-i38
      When 6348-7.3-1P-i39
      6348-7.3-1P-i40
      It implies,6348-7.3-1P-i41
      6348-7.3-1P-i42

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