How many numbers between 650 and 780 leave the remainders 4 and 6 when divided by 7 and 11, respectively?

Step-by-step explanation:

Divide by 7

The first thing you need to do is find the smallest member of this group. The first number you hit with this property is 655. Just divide each number from 651 to 655 until you hit the one that leaves a remainder of 4.

Now you can find the largest one. Start with 779 and work backwards. You will find that it is 774.

Every 7th number has this property.

d = 7

a = 655

L = 774

n = ?

L = a + (n - 1)*d

774 = 655 + (n - 1)*7

119 = (n - 1)*7

119/7 = 17

17 = n-1

n = 18

For 11

you follow the same procedure.

It turns out that n = 12

The wording is not clear enough to make a definitive call here if this is the right method. The question could mean that these two conditions must both be true. There won't be very many of them.

In fact there are just 2: 655 and 732.

I think this goes beyond what is reasonable to ask, and I don't think it is the answer, but I include it for completeness sake.