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  3. 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,

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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 = 1032 + 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 1032 + 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 101 + 2, 102+ 2, 103+ 2, 104+ 2, etc., are divided by 11 to see if there is a pattern that can help you determine the remainder when 1032+ 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 1032 + 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 102 + 2 and 104+ 2, respectively, so that is why the remainders are 3 when 102 + 2 and 104 + 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 1032 + 2, so the remainder is 3 when 1032 + 2 is divided by 11. Thus the correct answer is Choice C.
An alternative approach is to rewrite the expression 1032 + 2 using the factoring technique x2-1 =(x-1)(x+1)repeatedly, as follows.
1032 + 2 =(1032-l)+3
=(1016-l)(1016 + l)+3
=(108-l)(108 + l)(1016 + l)+3
=(104 -1)(104 +1)(108 + 1)(1016 +l)+3
=(102-l)(102+l)(104+l)(108+ 1)(1016+ l)+3
=(10+l)(10-l)(102+l)(104+l)(108+l)(1016+ l)+3
=11((10-l)(102+l)(104+l)(108+l)(1016+ l))+3
= 11k+3
Where k=(10-1)(102+1)(104+1)(108+ 1)(1016+ 1)is an integer. Since 1032+ 2 is of the form 11k + 3, where k is an integer, it follows that when 1032+ 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
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