Placement Exam For Level A - Summer High School Program, 2015 Page 4

2015 Summer High School Program

Placement Exam for Level A

NAME: ______________________

Long Answer: Write your solution in the space below each question. Make sure you include sufficient justification.

19. A number whose digits are all 1's is called a repunit. For example, 1, 11, and 111 are all repunits.

a. Is 111 prime? Is 1111 prime?

b. Can a repunit other than 11 with an even number of digits be a prime? Justify your answer.

20. Let �� ( �� ) denote how many positive integers with digits 1 and/or 2 have �� be the sum of their digits.

For example, �� ( 4 ) = 5 because there are 5 such integers (1111, 112, 121, 211, and 22)

a. What pattern is made by the values in the sequence �� ( 1 ) , �� ( 2 ) , �� ( 3 ) , �� ( 4 ) , �� ( 5 ) , … ?

b. Prove that the pattern holds forever.

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