BTGmoderatorDC wrote:Let S be the set of all positive integers having at most 4 digits and such that each of the digits is 0 or 1. What is the greatest prime factor of the sum of all the numbers in S ?
A. 11
B. 19
C. 37
D. 59
E. 101
Let S = the set of 4-digit integers between 0000 and 1111, inclusive, such that each digit is either 0 or 1.
Number of options for the thousands place = 2. (0 or 1)
Number of options for the hundreds place = 2. (0 or 1)
Number of options for the tens place = 2. (0 or 1)
Number of options for the units place = 2. (0 or 1)
To combine these options, we multiply:
2*2*2*2 = 16 integers
Since each of the 16 integers has 4 digits, there are 16 thousands digits, 16 hundreds digits, 16 tens digits, and 16 units digits.
Each digit has an equal chance of being 0 or 1.
Thus:
Eight of the 16 thousands digits will be 0, while the other eight will be 1, with the result that the sum for the thousands place = 8*0000 + 8*1000 = 8000
Eight of the 16 hundreds digits will be 0, while the other eight will be 1, with the result that the sum for the hundreds place = 8*000 + 8*100 = 800
Eight of the 16 tens digits will be 0, while the other eight will be 1, with the result that the sum for the tens place = 8*00 + 8*10 = 80
Eight of the 16 units digits will be 0, while the other eight will be 1, with the result that the sum for the units place = 8*0 + 8*1 = 8
Resulting sum:
8000 + 800 + 80 + 8 = 8888
8888 = 8 * 1111 = 8 * 11 * 101
Thus, the greatest prime factor for the sum = 101.
The correct answer is
E.
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
