Math Word Problem

A farmer is taking her eggs to market in her cart, but she hits a pot-hole, which knocks over all the containers of eggs. Though she herself is unhurt, every egg is broken. So she goes to her insurance agent, who asks how many eggs she had. She says she doesn't know, but she remembers some things from various ways she tried packing the eggs. She knows that when she put the eggs in groups of two, there was one left over. When she put the eggs in groups of three, there was also one egg left over. the same thing happened when she put them into groups of four, groups of five, and groups of six. But when she put them in groups of seven, she ended up with comlpete groups of seven with no eggs left over.

What is the smallest number of eggs the farmer had?

Explain how you solved this problem

the smallest number is 301.

the number you are looking for is 1 modulo 2, 1 modulo 3, and also 1 modulo 4 5 and 6, but 0 modulo 7. a number that is 1 mod 4 is automatically 1 mod 2, and a number that is 1 mod 6 is automatically 1 mod 3. So you can leave those conditions out. you now have to find a number x you can write as x=4k+1=5m+1=6n+1 for some k,m,n that are whole and positive numbers. so 4k=5m=6n. that means k is dividable by 5 and 3, m by 4 and 3 and n by 2 and 5. so x=4*5*3*k'+1 =5*4*3*m'+1 =6*2*5*n'+1. that means you can write x as 60*n'+1 and x is also dividable by 7. now you can just try. 61 is not good, 121 neither, and neither are 181 and 241. the next one is 301 and that is 43*7. 🙂 If someone doesn't understand this, just ask me and i'll try to explain it a bit more.

perhaps it's more clear if you say that 60 is the lcm (least common multiple) of 2,3,4,5,6 , and that x-1 must be a common multiple of them.

Originally posted by Lord Burn
A farmer is taking her eggs to market in her cart, but she hits a pot-hole, which knocks over all the containers of eggs. Though she herself is unhurt, every egg is broken. So she goes to her insurance agent, who asks how many eggs she had. She says she doesn't know, but she remembers some things from various ways she tried packing the eggs. She ...[text shortened]... r.

What is the smallest number of eggs the farmer had?

Explain how you solved this problem

The given states that the numbers of eggs the woman had is divisible by 2, 3, 4, 5 and 6 rest 1. Since 6 = 2*3 and 4= 2*2 the number has as divisors 2 (two times), 3 and 5.

The smallest number with that property is 2*2*3*5 + 1= 61
Unfortunately this number isn't divisible by 7. The general form of number divisible by 2, 2, 3 and 5 rest 1 is N*2*2*3*5 + 1.

The first few are thus 61, 121, 181, 241 and 301. The smallest of these wich is divisible by 7 is 301.

Thus 301 is the smallest number wich can be written as 2n+1, 3m+1, 4p+1, 5q+1, 6r+1 and 7s. With n, m, p, q, r and s certain positive integers.

Ton