Modulo Congruency

How to prove that $$7(100 ^{100}) + 8 \equiv 3^{90} - 21\ (\mathrm{mod} 28)?$$

I assumed the statement to be true. Then went on to show that both sides have the same remainder when divided by $28$.

$7(100)(100^{99}) + 8$ gives a remainder of $8$ when divided by $28$.

I don't know how to get the same remainder from $3^{90} - 21$.

• You have that $3^3 \equiv 27 \equiv -1 \mod 28$. Use this to show that $3^{90} \equiv 1 \mod 28$. Are you sure that the LHS is equivalent with 8 mod 28? – Student Feb 24 '17 at 17:57

You know that $$3^3\equiv 97\equiv -1\pmod{28}.$$

So

$$3^{90}-21\equiv (-1)^{30}-21\equiv 8\pmod{28}.$$

And you can check that $$100^4\equiv 100\pmod {28}$$

so

$$7(100^{100})+8\equiv 7\times 100+8\equiv 8\pmod{28}.$$

So this statement is true.

• You have made an error: $3^3 = 27$... – Student Feb 24 '17 at 17:59
• @Student You are right :) I edited since. – E. Joseph Feb 24 '17 at 17:59

to calculate the remainder of $3^{90}-21$ you should use eulers theorem on the left side, notice $\varphi(28)=12$, so $3^{90}=3^{84}3^6\equiv 3^6\equiv 27^2\equiv 1 \bmod 28$, Hence $3^{90}-21\equiv 1-21\equiv-20\equiv 8\bmod 28$

As you have observed, $7\cdot 100^{100} \equiv 0 \bmod 28$. So what we want to prove then is $3^{90}\equiv 8+21\equiv29\equiv 1 \bmod 28$

The Carmichael function $\lambda(28)={\rm lcm}(\lambda(4),\lambda(7)) = {\rm lcm}(2, 6) = 6$ means that the cycle of values for $3^k \bmod 28$ is either $6$ in length or divides $6$ in length, and since $\gcd(3,28)=1$ then we know $3^6\equiv 1 \bmod 28$. In fact we can also see this directly since $3^3=27\equiv-1\bmod 28$ (but this discovery is not needed).

So since $3^6\equiv 1 \bmod 28$ then also $3^{90}\equiv (3^6)^{15}\equiv 1^{15}\equiv 1\bmod 28$ and the result holds.