Suppose that a and b are integers, a≡11(mod19), and b≡3(mod19). Find the integer c with 0 <= c <= 18 such that c≡a^3+4b^3(mod19)

Arnav Heath

Arnav Heath

Open question

2022-08-19

Suppose that a and b are integers, a11(bmod19), and b3(bmod19). Find the integer c with 0c18 such that ca3+4b3(bmod19)

Answer & Explanation

Carmelo Peck

Carmelo Peck

Beginner2022-08-20Added 12 answers

Definitions
Division algorithm Let a be an integer and d a positive integer. Then there are unique integers q and r with 0r<d such that a=dq+r
q is called the quotient and r is called the remainder
q=a ÷ d
r=abmodd
Theorem 5 Let m be a positive integer. If ab(bmodm) and cd(bmodm), then a+cb+d(bmodm) and acbd(bmodm).
Solution
a=11(bmod19)
b3(bmod19)
0c18
Use theorem 5:
ca3+4b3(bmod19)
=113+433(bmod19)
=1331+427(bmod19)
=1331+108(bmod19)
=1439(bmod19)
=14(bmod19)
We then obtain c=14 with 0c18.

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