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:
Mo2men wrote:In the first 1000 positive integers, how many integers exist such that they leave a remainder 4 when divided by 7, and a remainder 9 when divided by 11?
A. 10
B. 11
C. 12
D. 13
E. 14
When x is divided by 7, the remainder is 4.
x = 7a + 4 = 4, 11, 18, 25, 32, 39, 46,
53...
When x is divided by 11, the remainder is 9.
x = 11b + 9 = 9, 20, 31, 42,
53...
Thus, when x is divided by 77 -- the LCM of 7 and 11 -- the remainder will be 53 (the smallest value common to both lists).
x = 77c + 53 = 53, 130, 207...
In the resulting list, the smallest value is yielded when c=0:
77*0 + 53 = 53.
The greatest value less than 1001 is yielded when c=12:
77*12 + 53 = 977.
Implication:
c can be any integer between 0 and 12, inclusive, yielding a total of 13 options.
The correct answer is
D.
Private tutor exclusively for the GMAT and GRE, with over 20 years of experience.
Followed here and elsewhere by over 1900 test-takers.
I have worked with students based in the US, Australia, Taiwan, China, Tajikistan, Kuwait, Saudi Arabia -- a long list of countries.
My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
I unlock the best way for YOU to solve problems.
For more information, please email me (Mitch Hunt) at
[email protected].
Student Review #1
Student Review #2
Student Review #3
