A quick lesson on remainders:
When x is divided by 5, the remainder is 3.
In other words, x is 3 more than a multiple of 5:
x = 5a + 3.
When x is divided by 7, the remainder is 4.
In other words, x is 4 more than a multiple of 7:
x = 7b + 4.
Combined, the statements above imply that when x is divided by 35 -- the LOWEST COMMON MULTIPLE OF 5 AND 7 -- there will be a constant remainder R.
Put another way,
x is R more than a multiple of 35:
x = 35c + R.
To determine the value of R:
Make a list of values that satisfy the first statement:
When x is divided by 5, the remainder is 3.
x = 5a + 3 = 3, 8, 13,
18...
Make a list of values that satisfy the second statement:
When x is divided by 7, the remainder is 4.
x = 7b + 4 = 4, 11,
18...
The value of R is the SMALLEST VALUE COMMON TO BOTH LISTS:
R = 18.
Putting it all together:
x = 35c + 18.
Another example:
When x is divided by 3, the remainder is 1.
x = 3a + 1 = 1, 4, 7, 10,
13...
When x is divided by 11, the remainder is 2.
x = 11b + 2 = 2,
13...
Thus, when x is divided by 33 -- the LCM of 3 and 11 -- the remainder will be 13 (the smallest value common to both lists).
x = 33c + 13 = 13, 46, 79...
Onto the problem at hand:
When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
When n is divided by 13, the remainder is 2.
n = 13a + 2 = 2, 15, 28, 41, 54, 67, 80,
93...
When n is divided by 8, the remainder is 5.
n = 8b + 5 = 5, 13, 21, 29, 37, 45. 53, 61, 69, 77, 85,
93...
Thus, when n is divided by 104 -- the LCM of 13 and 8 -- the remainder will be 93 (the smallest value common to both lists).
n = 104c + 93 = 93, 197...
Of the list of options for n, only the first -- 93 -- is less than 180.
The correct answer is
B.
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