题目信息
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
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正确答案：
E:101
Arithmetic Properties of integers
By writing down all the positive integers in S, their sum can be found.
The sum of these integers is 8,888. Since this sum is 8 × 1,111 = 23 × 11 × 101 (note that 1,111 = (11 × 100) + 11), it follows that 101 is the largest prime factor of the sum.
Alternatively, we can simplify the description by letting the integers having fewer than four digits be represented by four-digit strings in which one or more of the initial digits is 0. For example, the two-digit number 10 can be written as 0010 = (0 × 103) + (0 × 102) + (1 × 101) + (0 × 100). Also, we can include 0 = 0000, since the inclusion of 0 will not affect the sum. With these changes, it follows from the Multiplication Principle that there are 24 = 16 integers to be added. Moreover, for each digit position (units place, tens place, etc.) exactly half of the integers will have a digit of 1 in that digit position. Therefore, the sum of the 16 integers will be (8 × 103) + (8 × 102) + (8 × 101) + (8 × 100), or 8,888. Note that this alternative method of finding the sum is much quicker than the other method if “at most four digits” had been “at most seven digits.” In the case of “at most seven digits,” there will be 27 = 128 integers altogether, and for each digit position, half of the integers will have a digit of 1 in that digit position and the other half will have a digit of 0 in that digit position. Thus, the sum will be (64 × 106) + (64 × 105) + … + (64 × 100) = 71,111,104. Incidentally, finding the greatest prime factor of 71,111,104 is not appropriate for a GMAT problem, but in this case a different question about the sum could have been asked.
The correct answer is E.
By writing down all the positive integers in S, their sum can be found.
1 |
10 |
11 |
100 |
101 |
110 |
111 |
1,000 |
1,001 |
1,010 |
1,011 |
1,100 |
1,101 |
1,110 |
1,111 |
The sum of these integers is 8,888. Since this sum is 8 × 1,111 = 23 × 11 × 101 (note that 1,111 = (11 × 100) + 11), it follows that 101 is the largest prime factor of the sum.
Alternatively, we can simplify the description by letting the integers having fewer than four digits be represented by four-digit strings in which one or more of the initial digits is 0. For example, the two-digit number 10 can be written as 0010 = (0 × 103) + (0 × 102) + (1 × 101) + (0 × 100). Also, we can include 0 = 0000, since the inclusion of 0 will not affect the sum. With these changes, it follows from the Multiplication Principle that there are 24 = 16 integers to be added. Moreover, for each digit position (units place, tens place, etc.) exactly half of the integers will have a digit of 1 in that digit position. Therefore, the sum of the 16 integers will be (8 × 103) + (8 × 102) + (8 × 101) + (8 × 100), or 8,888. Note that this alternative method of finding the sum is much quicker than the other method if “at most four digits” had been “at most seven digits.” In the case of “at most seven digits,” there will be 27 = 128 integers altogether, and for each digit position, half of the integers will have a digit of 1 in that digit position and the other half will have a digit of 0 in that digit position. Thus, the sum will be (64 × 106) + (64 × 105) + … + (64 × 100) = 71,111,104. Incidentally, finding the greatest prime factor of 71,111,104 is not appropriate for a GMAT problem, but in this case a different question about the sum could have been asked.
The correct answer is E.
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