SELECTED PROBLEMS FROM HW6
4.1 16: Which decimal digits occur as the final digit of the fourth power of
an integer?
Note that if two positive integers are congruent mod 10 they have the same
last digit(and that the fourth power of an integer is always positive). For
any integer a, there is an integer d between 1 and 10 such that a ≡ d (mod
10). Then a4 ≡ d4 (mod 10), so the last digit of a4 will be equal to the last
digit of d4 . Computing d4 for d = 1, . . . , 10 we find that the last digits that
occur are 0, 1, 5, 6.
4.2 12: Show that if a is an inverse of a mod m, and b is an inverse of b mod
m, then ab is an inverse of ab mod m.
We have aa ≡ 1 mod m since a is an inverse of a. Similarly bb ≡ 1 mod m.
We must show that abab ≡ 1 mod m. But
abab = aabb ≡ 1 ∗ 1 ≡ 1 (mod m).
4.3 18: Does the system of congruences x ≡ 1 (mod 8), x ≡ 3 (mod 9),
x ≡ 2 (mod 12) have any simultaneous solutions?
No. If x ≡ 1 (mod 8) then x is odd, but if x ≡ 2 (mod 12) then x is even.
Therefore both can’t happen at the same time.
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