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Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given,
Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given,
admin
2015-03-24
24
问题
Compare Quantity A and Quantity B, using additional information centered above the two quantities if such information is given, and select one of the following four answer choices:
(A)Quantity A is greater.
(B)Quantity B is greater.
(C)The two quantities are equal.
(D)The relationship cannot be determined from the information given.
A symbol that appears more than once in a question has the same meaning throughout the question.
m = 10
32
+ 2
When m is divided by 11, the remainder is r.
Quantity A Quantity B
r 3
选项
答案
C
解析
Actually dividing 10
32
+ 2 by 11 would be very time consuming, so it is worth trying to compare the quantities without actually doing the division.
A good approach would be to compute the remainders when 10
1
+ 2, 10
2
+ 2, 10
3
+ 2, 10
4
+ 2, etc., are divided by 11 to see if there is a pattern that can help you determine the remainder when 10
32
+ 2 is divided by 11. The following table shows the first few cases.
Note that the remainder is 1 when 10 is raised to an odd power, and the remainder is 3 when 10 is raised to an even power. This pattern suggests that since 32 is even, the remainder when 10
32
+ 2 is divided by 11 is 3.
To see that this is true, note that the integers 99 and 9,999 in the rows for n = 2 and n = 4, respectively, are multiples of 11. That is because they each consist of an even number of consecutive digits of 9. Also, these multiples of 11 are each 3 less than 10
2
+ 2 and 10
4
+ 2, respectively, so that is why the remainders are 3 when 10
2
+ 2 and 10
4
+ 2 are divided by 11. Similarly, for n = 32, the integer with 32 consecutive digits of 9 is a multiple of 11 because 32 is even. Also, that multiple of 11 is 3 less than 10
32
+ 2, so the remainder is 3 when 10
32
+ 2 is divided by 11. Thus the correct answer is Choice C.
An alternative approach is to rewrite the expression 10
32
+ 2 using the factoring technique x
2
-1 =(x-1)(x+1)repeatedly, as follows.
10
32
+ 2 =(10
32
-l)+3
=(10
16
-l)(10
16
+ l)+3
=(10
8
-l)(10
8
+ l)(10
16
+ l)+3
=(10
4
-1)(10
4
+1)(10
8
+ 1)(10
16
+l)+3
=(10
2
-l)(10
2
+l)(10
4
+l)(10
8
+ 1)(10
16
+ l)+3
=(10+l)(10-l)(10
2
+l)(10
4
+l)(10
8
+l)(10
16
+ l)+3
=11((10-l)(10
2
+l)(10
4
+l)(10
8
+l)(10
16
+ l))+3
= 11k+3
Where k=(10-1)(10
2
+1)(10
4
+1)(10
8
+ 1)(10
16
+ 1)is an integer. Since 10
32
+ 2 is of the form 11k + 3, where k is an integer, it follows that when 10
32
+ 2 is divided by 11, the remainder is 3. The correct answer is Choice C.
This explanation uses the following strategies.
Strategy 5: Simplify an Arithmetic or Algebraic Representation
Strategy 7: Find a Pattern
Strategy 11: Divide into Cases
转载请注明原文地址:https://kaotiyun.com/show/NwkO777K
本试题收录于:
GRE QUANTITATIVE题库GRE分类
0
GRE QUANTITATIVE
GRE
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