$N$ is a positive integer. When $N$ is divided by $3$, the remainder is $2$. When $N$ is divided by $4$, the remainder is $3$. What is the remainder when $N$ is divided by $12$?

When 482 , 992 , 1094are divided by a positive integer D, the remainders are all 74. What is D?

N is a four-digit positive integer. Dividing N by 9 , the remainder is 5. Dividing N by 7, the remainder is 3. Dividing N by 5, the remainder is 1. What is the smallest possible value of N?

A box contains many candies.

If 3 candies are taken away, the rest can be shared evenly among 7 people.

If 6 are taken away, the rest can be shared evenly among 8 people.

If 3 candies are added, then the box can be shared evenly among 5 people.

What is the smallest number of candies in the box?

Guest Sep 26, 2020

#1**0 **

I love these problems!

1) N=**11 when divided by 12**

So if you are familiar with modular arithmetic, we will write some equations.

N==2 (mod 3)

N==3 (mod 4)

N==x(mod 12)

Let's solve for N.

3a+2=4b+3

a and b are integers

take the whole thing modulo 3

b=2(mod 3)

so inputting for a new quotient integer c,

N=3a+2=4(3c+2)+3

N=3a+2=12c+11

N= **11** (mod 12)

or the remainder is 11 when divided by 12

2) D=**102**

482-74=408

992-74=918

1094-74=1020

The GCF is 102.

D=**102**

!!!

3) N=**311**

Again, use modular arithmetic with integer variable quotients. Solving the way I did before, N=311 (mod 315) so N>=311+315n, and n can't be negative thus N=**311.**

4) **38 candies**

lol, same modular arithmetic.

Solving equations and writing them, we have for the # of candies=x

x=3(mod7)

x=6(mod8)

x=3(mod5)

so x=3(mod 35)

and x=6(mod 8)

therefore x= 38 mod 280, and x=**38 **

**(because 318 would obviously be too many candies for >8 people to eat.)**

Pangolin14 Sep 26, 2020