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A) If $a\equiv 62\pmod{99}$ and $b\equiv 75\pmod{99}$, then for what integer $n$ in the set $\{1000,1001,1002,\ldots,1097,1098\}$ is it true that $$a-b\equiv n\pmod{99}~?$$

**I have edited so that it can be read. Next time edit it yourself. Melody.**

**(It still doesn't make sense though)**

\(\color{blue} {If\;\;a\equiv 62\pmod{99} \;\;and \;\; b\equiv 75\pmod{99}\\ \text{ then for what integer n in the set } \{1000,1001,1002,\ldots,1097,1098\}\\\text{ is it true that }a-b\equiv n\pmod{99}~?}\)

B) The remainders when three positive integers are divided by 5 are 1, 2, and 3. Find the remainder when their product is divided by 5.

C) What is the residue modulo of the sum of the modulo inverses of the first positive integers?

Express your answer as an integer from to , inclusive.

D) Solve the congruence , as a residue modulo 43. (Give an answer between 0 and 42.)

Guest Dec 8, 2018

#1**0 **

a) Assume a = 62 and b = 75. a - b = -13, which equals 86 (mod 99).

The problem is asking n \(\cong\) 86 (mod 99) for the number \(n\) in the set {1000, 1001....1098}.

We can clearly see that 990 (mod 99) = 0. So 990 + 99 = 1089, which is also divisible by 99. So if n = 1089, then \(n \cong 0 (mod99)\). But we need \(n \cong 86 (mod99)\), so n = 1089 - 13 = 1076. The anwer is \(\boxed{1076}\).

Hope this helps,

- PM

PartialMathematician Dec 9, 2018

#3**0 **

b) Just plug in numbers 1, 2, and 3.

1*2*3 = 6. 6/5 = 1 R1.

Remainder \(\boxed{1}\).

- PM

PartialMathematician Dec 9, 2018

#5**0 **

I think you messed up on the LaTeX in C and D. I cant read it...

sorry,

- Pm

PartialMathematician Dec 9, 2018