Question 363643
I'm not sure that a child will like a general method, but it is
rather useful in math competitions. Please review a chapter about
linear Diophantine equations in any math textbook.

N = 5a + 2 = 7b + 4

where a and b are some natural numbers. Let's solve this
Diophantine equation.

5a - 7b = 2
5(a - b) = 2(b + 1)

This can happen when
a - b = 2
b + 1 = 5

i.e. 

a = 6 
b = 4
N = 32

Addition of LCM(7,5)=35 won't change the remainders.
So the general solution is
N = 32 + 35*n where n is a natural number.

Now we need to solve an inequality:
40 < 32 + 35*n < 80

i.e.
n = 1
N = 67