$N$ 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?
+421 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.)
