Question

An old woman goes to market and a horse steps on her basket and crushes all the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time there were none left over. What is the smallest number of eggs she could have had?

Answer 1

Let \(x\) = number of eggs in her basket

We need to find the minimum positive integer, \(x\), that satisfies :
\[
x = 2a+1\\
x = 3b+1\\
x = 4c+1\\
x = 5d+1\\
x = 6e+1\\
x = 7f~~~~~~
\]

Answer 2

should the one for 7 also have a +1?

Answer 3

nope, because `but when she took them seven at a time there were none left over. `

Answer 4

oh okay! so when I'm doing the math to figure all of this out, am I doing 2*1+1 and so on to see which ones do not give me remainders??

Answer 5

You have the right idea which works but that would be very painful

what if the oldie has million eggs in her basket ?
would you be testing all million numbers ?

Answer 6

noooo lol

Answer 7

Look at this number :
\[n = 2*3*4*5*6\]

whats the remainder when \(n\) is divided by 2 ?

Answer 8

0!

Answer 9

and what about the remainders when \(n\) is divided by 3, 4, 5 and 6

Answer 10

0 for all of those too!

Answer 11

good, before it gets boring
can you find a smaller positive number that is exactly divisible by 2, 3, 4, 5 and 6 ?

Answer 12

90??

Answer 13

let me rephrase the question :
what is the "least" positive number that is divisible by 2, 3, 4, 5 and 6 ?

Answer 14

so you're telling me she had 90 eggs in that basket?

Answer 15

not at all

Answer 16

wait never mind 90 is divisible by all of them except 4

Answer 17

btw the least number is not 90
you can do better, try again

Answer 18

here is a hint : "lcm"

Answer 19

13?!

Answer 20

that fails when u divide by 5

Answer 21

wait no because there is R1

Answer 22

hm...

Answer 23

whats the lcm of 2,3,4,5,6 ?

Answer 24

2

Answer 25

Okay thats wrong
Never heard of "lcm" and "gcd" before ?

Answer 26

would lcm be 720?

Answer 27

or 60?

Answer 28

60 is the lcm, yes

im pretty sure you knew these lcm and gcd stuff

Answer 29

haha yeah, I'm just thinking way too into everything now sorry about that!

Answer 30

its fine

so if the oldie has 60 eggs in her basket, then if she takes out 2 eggs at a time, 0 eggs will be left in the end.

but we want 1 egg in the end, so let the number of eggs be 60+1 = 61 and see if it works

Answer 31

what are the remainders when you divide 61 by 2, 3, 4, 5, and 6 ?

Answer 32

i got R1 for all except 4, 4 was R5

Answer 33

are u really sure ?

Answer 34

hold on!

Answer 35

whah messed up big time there! theyre all R1

Answer 36

good so \(61\) satisfies all the equations except the last one
\[x=7f\]

Answer 37

so then there are 61 eggs!

Answer 38

or do i need to adjust my answer since 61/7 isn't a perfect fit?

Answer 39

Exactly! we need to adjust it so that last equation is also satisfied

Answer 40

Notice that any number of form \(60k+1\) leaves the remainder \(1\) when divided by 2, 3, 4, 5, or 6

Answer 41

plugin k=1, 2, 3 till you get a number thats divisible by 7

Answer 42

k=2 :
60(2) + 1 = 121 not divisible by 7

k=3 :
60(3) + 1 = 181 not divisible by 7

k=4 :
60(4) + 1 : 241 not divisible by 7

k=5 :
60(5) + 1 : 301 divisible by 7

so the least possible eggs that the old woman could be having is 301

Answer 43

5!!

Answer 44

sorry, it took me a little while but i actually did get that answer!

Answer 45

thank you so much for your help and especially your patience!! i really appreciate it :)

Answer 46

np:)