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$N$ 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?

 Sep 26, 2020
 #1
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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.)

 Sep 26, 2020
 #2
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Thank you so much!

 Sep 26, 2020

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