How many integers between 57 and 70, inclusive, have a remainder of 2 when divided by 7?

A number divisible by 7 can be expressed as 7n, where n is a positive integer.
A number has a remainder of 2 when divided by 7 can be expressed as 7n + 2.

The number is between 57 and 70:
57 ≤ 7n + 2 ≤ 70
57 - 2 ≤ 7n + 2 -2 ≤ 70 - 2
55 ≤ 7n ≤ 68
55/7 ≤ 7n/7 ≤ 68/7
7.86 ≤ n ≤ 9.7

When n = 8: 7n + 2 = 58
When n = 9: 7n + 2 = 65

Answer: There are 2 integers, i.e. 58 and 65.

57 < 7n + 2 < 70
55 < 7n < 68
55/7 < n < 68/7
7.86 < n < 9.70

n = 8 , 9

7 * 8 + 2 = 56 + 2 = 58
7 * 9 + 2 = 63 + 2 = 65

Let 57=<x<=70 &
x=2(mod 7)=>
x=2+7k, where k=0,+/-(1,2,3,...)
k=8=>x=58
k=9=>x=65
Thus, 58, 65 are the 2 integers required.

The easiest way I can think of is to take all the multiples of 7 that are near that range and add 2. Then see how many are between 57 and 70. 

7*8=56+2=58
7*9=63+2=65
7*10=70+2=72 (this one doesn’t work)
So the only ones are 58 & 65. The answer is 2. 

56 & 63 are divisible by. 58 & 65 both have a remainder of 2 when divided by.
All numbers having a remainder of 2 when divided by 7 are 2 greater than any
number divisible by 7. Suppose you were required to find all numbers between
200 and 400, ie., greater than 200 & less than 400 which have a remainder of 2
when divided by 7. We first determine that 203 (= 7*29) is the first number in the 
range that is divisible by 7. So the first number in the range with a remainder of
2 when divided by 7 is 205. We set up an AP (arithmetic progression) of form
a, a+d, a+2d, ......,a+(n-1)d,... = t(1), t(2), t(3),....,t(n),...where t(n) = a+(n-1)d for
n = 1,2,3,.....We set a = 205, d = 7 and require t(n) is less that 400...(1).
(1) implies 205+(n-1)(7) less than 400, ie., n less than 1+(400-205)/7 , ie., n less than 28.8571428. Our progression therefore contains 28 terms satisfying all conditions.
 

Integers between 57 and 70, inclusive, 
which have a remainder of 2 when divided by 7:
58, and 65

There are only two; they are:-
58 = 8*7 + 2 and
65 = 9*7 + 2

Two numbers: 58 and 65

There are 12 numbers in that range.  Just try them all. That's a sure-fire technique.