What is the smallest number of eggs in a basket if one egg i

What is the smallest number of eggs in a basket if one egg is left over when the eggs are removed 2,3, or 5 at a time, but no eggs are left over when they are removed 7 at a time.

Solution

So,

x=1 mod 2

x=1 mod 3

x=1 mod 5

x=0 mod 7

So, x=2m+1,m=0,1,2,3,

Since, x=1 mod 3, we must have m=0 mod 3 ie m must be a multiple of 3 ie m=3n,n=0,1,2,3

x=6n+1

6n+1=1 mod 5

hence, 6n=0 mod 5

Hence, n=0 mod 5

Hence, n=5k,k=0,1,2,3,

So, x=30k+1

Let us write them down and see what is smallest number so that:x=0 mod 7

1,31,61,91

We see 91 is the smallest number\\

So smallest number of eggs is 91