Strategy 7: Find a Pattern

Patterns are found throughout mathematics. Identifying a pattern is often the first step in understanding a complex mathematical situation. Pattern recognition yields insight that may point in the direction of a complete solution to the problem or simply help you generate a hypothesis, which requires further exploration using another strategy. In a problem where you suspect there is a pattern but don’t recognize it yet, working with particular instances can help you identify the pattern. Once a pattern is identified, it can be used to answer questions.

• This strategy is used in the following two sample questions.

This is a Multiple-Choice – Select One or More Answer Choices Question.

  1. Which of the following could be the units digit of 57 to the power n where n is a positive integer?

    Indicate all such digits.

    (A) 0
    (B) 1
    (C) 2
    (D) 3
    (E) 4
    (F) 5
    (G) 6
    (H) 7
    (I) 8
    (J) 9


    The units digit of 57 to the power n is the same as the units digit of 7 to the power n for all positive integers n. To see why this is true for n is equal to 2 compute 57 to the power 2 by hand and observe how its units digit results from the units digit of 7 to the power 2 Because this is true for every positive integer n, you need to consider only powers of 7. Beginning with n is equal to 1 and proceeding consecutively, the units digits of 7, 7 to the power 2 7 to the power 3 7 to the power 4 and 7 to the power 5 are 7, 9, 3, 1 and 7, respectively. In this sequence, the first digit, 7, appears again, and the pattern of four digits, 7, 9, 3, 1, repeats without end. Hence, these four digits are the only possible units digits of 7 to the power nand therefore of 57 to the power n The correct answer consists of Choices B (1), D (3), H (7) and J (9).

    Note: This question and explanation also appear as an example of Strategy 12.

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