Find the smallest amount of eggs that can be in a basket if

Find the smallest amount of eggs that can be in a basket if: When the eggs are removed 2,3,4,5, and 6 at a time, there are 1,2,3,4, and 5 eggs respectively, left over, and when the eggs are removed 7 at a time, there are none left over.

What is the least possible amount of golden coins that the thieves stole?

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**online solutions to this problem are kind of vague with a lot of shortcuts and hidden steps of solution. Is there only one solution to the problem?

or there can be many?

Solution

Here as 7 at a time removed, there are non left over.

That means if total eggs be x, then x= 0 (mod 7)

or x= 7a ..........(i) (where a is any integer)

Further based on first equations,

x= 1(mod 2) i.e. x= 1+2b .........(ii)

also x= 2( mod 3) i.e. x= 2+ 3c .........(iii)

again x= 3(mod 4) i.e. x= 3+ 4d ........(iv)

and x= 4 (mod 5) i.e. x= 4+ 5e ........(v)

and also x= 5(mod 6) i.e. x= 5+ 6f .........(vi)

Now we have using substitution,

x=1+2a = 2+3b= 3+4c = 4+5d = 5+6 e = 7f

and this condition is being satisfied when x= 119

For example 119 = 1 + 2( 59) = 2+ 3(39) = 3+ 4(29) = 4+ 5(23) = 5+ 6(114) = 7(17)

So smallest amount of eggs = 119

Answer