N is a four-digit positive integer. Dividing N by 9, the remainder is 5. Dividing N by 7, the remainder is 4. Dividing N by 5, the remainder is 1. What is the smallest possible value of N?

Guest Apr 20, 2021

#1**0 **

The smallest 4-digit n =1256

1256 mod 9 =5

1256 mod 7 =3

1256 mod 5 =1

XxMysteriousGirlxX Apr 20, 2021

#3**+1 **

How did you arrive at this answer Hannah?

I do not understand your working either guest. What is CRT + MMI ?

Melody
Apr 20, 2021

#4**+1 **

I'm not quite sure how Hannah or Guest got their answers, but I think there's a theorem called Chinese remainder theorem that you can use.

x = 5 (mod 9)

x = 3 (mod 7)

x = 1 (mod 5)

To use the chinese remainder therum, the gcd of all mods much be 1.

gcd(9, 7) = 1

gcd(7, 5) = 1

gcd(9, 5) = 1

It's a bit hard to explain, but there are good videos on it on yt. :))

One by "Randell Heyman."

Are mods are 9, 7, and 5.

(not 9) (not 7) (not 5)

x = (7*5) + (9*5) + (9*7)

First part (9 mod)

Now, it gets a bit confusing.

(9*5) and (9*7) will both be 0 in mod 9 (multiple of 9)

So we need to find a y*7*5 = 5 in mod 9.

y = 4.

Second part (7 mod)

x = (4*7*5) + (9*5) + (9*7)

(4*7*5) and (9*7) will both be 0 in mod 7.

(y*9*5) = 3 in mod 7.

y = 1, 45 = 3 in mod 7

Third part (5 mod)

x = (4*7*5) + (9*5) + (9*7)

(4*7*5) and (9*5) will both be 0 in mod 5.

(9*7*y) = 1 in mod 5

y = 2

Final equation

x = (4*7*5) + (9*5) + (9*7*2) = 311

Yayyy, it works. :))

I learned smth new today.

However, we need a 4 digit number.

311+(9*7*5)y = smallest 4 digit number

x = 1256

That has to be the most terrible explanation ever... but I tried. :)

=^._.^=

catmg
Apr 20, 2021

#5**+1 **

Read the question carefully!

N mod 9 ==5

N mod 7 ==4 [NOT 3]

N mod 5 ==1

See this link here: https://web2.0calc.com/questions/congruences

Guest Apr 20, 2021

edited by
Guest
Apr 20, 2021

#7**+2 **

It doesn't matter that you copied the question incorrectly catmg. It is not YOUR homework.

You explained your __method__ the best you could and that is all that counts.

The asker is always expected to check then accept, edit or reject the answers given to them.

I learned from your answer in that I looked up the Chinese remainder theorem. Thanks

This is the video that I learned from. I found it excellent.

It is from a channel called "Maths with Jay"

I got the answer 1166 (which is the same as our first guest's answer.)

Melody
Apr 21, 2021