If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder, respectively, such that y= divisor * quotient + remainder = xq + r; and 0<=r < x.
For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since 15 = 6*2+3.
Notice that 0<= r < x means that remainder is a non-negative integer and always less than divisor.
This formula can also be written as y/x = q + r/x.
When y is divided by x the remainder is0 if y is a multiple of x.
For example, 12 divided by 3 yields the remainder of 0 since 12 is a multiple of 3 and 12 = 3*4+0.
When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer.
For example, 7 divided by 11 has the quotient 0 and the remainder 7 since 7=11*0+7
The possible remainders when positive integer y is divided by positive integer x can range from 0 to x-1.
For example, possible remainders when positive integer y is divided by 5 can range from 0 (when y is a multiple of 5) to 4 (when y is one less than a multiple of 5).
If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.
For example, 123 divided by 10 has the remainder 3 and 123 divided by 100 has the remainder of 23.
1. Collection of Methods
A way that the GMAT will test remainders is what you would typically just divide back into the problem to determine the decimals:
25/4 = 6 remainder 1
Divide that 1 back by 4 to get 0.25, so the answer is 6.25.
Any number with a remainder could be expressed as a decimal.
The remainder provides the data after the decimal point, and the quotient gives you the number to the left of the decimal point.
Consider this problem (which appears courtesy of GMAC):
Example： When positive integer x is divided by positive integer y, the remainder is 9. If x/y = 96.12,what is the value of y?
(A) 96 (B) 75 (C) 48 (D) 25 (E) 12
Going back to the concept of the remainder, the remainder of 9 is what will give us that 0.12 after the decimal place. The answer to the division problem x/y is either:
96 remainder 9
Therefore, when the remainder of 9 is divided back over y, we get 0.12. Mathematically, this means that:
9/y = 0.12
0.12y = 9
12y = 900
y = 900/12
y = 300/4
y = 75
The correct answer is B.
Given that an integer "n" when divided by an integer "a" gives "r" as reminder then the integer"n" can be written as
n = ak + r
where k is a constant integer.
Example 1： What is the remainder when B is divided by 6 if B is a positive integer?
(1) When B is divided by 18, the remainder is 3
(2) When B is divided by 12, the remainder is 9
STAT1 : When B is divided by 18, the remainder is 3
So, we can write B as
B = 18k + 3
Now, to check the reminder when B is divided by 6, we essentially need to check the reminder when 18k + 3 is divided by 6
18k goes with 6 so the reminder will 3
So, it is sufficient
STAT2 : When B is divided by 12, the remainder is 9
So, we can write B as
B = 12k + 9
Now, to check the reminder when B is divided by 6, we essentially need to check the reminder when 12k + 9 is divided by 6
12k goes with 6 so the remainder will be the same as the reminder for 9 divided by 6 which is 3
So, reminder is 3
So, it is sufficient.
Answer will be D
What is the remainder when positive integer t is divided by 5?
(1) When t is divided by 4, the remainder is 1
(2) When t is divided by 3, the remainder is 1
STAT1: When t is divided by 4, the remainder is 1
t = 4k +1
possible values of t are 1,5,9,13
Clearly we cannot find a unique reminder when t isdivided by 5 as in some cases(t=1) we are getting the reminder as 1 and insome(t=5) we are getting the reminder as 0.
STAT2: When t is divided by 3, the remainder is 1
t = 3s + 1
possible values of t are 1,4,7,10,13,16,19
Clearly we cannot find a unique reminder when t is divided by 5 as in some cases(t=1) we are getting the reminder as 1 and in some(t=10) we are getting the reminder as 0.