# Expert Answers

1. Explain your answers to the following questions. Today is Wednesday.

(a) What day will it be in 7 days?

(b) In 67 days what day will it be?

(c) If a government proposal were passed to lengthen the weekend by sticking in an extra day after Saturday, what day would it be 67 days from now?

2. Examine the number 4. 51283512835128351283 . . .

What is the 295th digit after the decimal point in this number? Explain how you got your answer.

3. A simple two-person game is as follows. There is a pile of pennies between you and your opponent. The two of you take turns taking either one, two, or three pennies. Your goal is to force your opponent to take the last penny. Try out the game with a friend, then answer the following questions.

(a) Fill in the missing entries of the following table.

(b) Let n be the number of pennies in the table. It is your turn. How many pennies should you take? Give the answer in terms of n.

(c) Repeat Problem a. for the variant of the game in which each player can take one, two, three, or four pennies:

(d) Repeat Problem b. for the variant.

4. For each of the following congruences, find the solution for x. Your solution should be a representative (one of 0, 1, 2, . . . , m − 1 where m is the modulus).

(a) x ≡ 4 + 3 (mod 8)

(b) x ≡ 7 + 34 (mod 4)

(c) 13 + 22 ≡ x (mod 13)

(d) 6 − 7 ≡ x (mod 88)

(e) x ≡ 9 − 19 (mod 7)

5. Find the (representative) solution for x.

(a) 12 + 4 + 7 + 12 + 17 ≡ x (mod 12)

(b) 7 + x ≡ 2 (mod 3)

(c) x ≡ 16 + 2 (mod 8)

(d) x ≡ 57 · 73 (mod 17)

(e) x ≡ 3 · (87 − 5) (mod 7)

(f) x ≡ (−5) − (−11) (mod 10)

6. Compute the following:

(a) 16 + 2 rem 8

(b) 57 · 73 rem 17

(c) 3 · (87 − 5) rem 7

(d) (−5) − (−11) rem 10

(a) What day will it be in 7 days?

(b) In 67 days what day will it be?

(c) If a government proposal were passed to lengthen the weekend by sticking in an extra day after Saturday, what day would it be 67 days from now?

2. Examine the number 4. 51283512835128351283 . . .

What is the 295th digit after the decimal point in this number? Explain how you got your answer.

3. A simple two-person game is as follows. There is a pile of pennies between you and your opponent. The two of you take turns taking either one, two, or three pennies. Your goal is to force your opponent to take the last penny. Try out the game with a friend, then answer the following questions.

(a) Fill in the missing entries of the following table.

(b) Let n be the number of pennies in the table. It is your turn. How many pennies should you take? Give the answer in terms of n.

(c) Repeat Problem a. for the variant of the game in which each player can take one, two, three, or four pennies:

(d) Repeat Problem b. for the variant.

4. For each of the following congruences, find the solution for x. Your solution should be a representative (one of 0, 1, 2, . . . , m − 1 where m is the modulus).

(a) x ≡ 4 + 3 (mod 8)

(b) x ≡ 7 + 34 (mod 4)

(c) 13 + 22 ≡ x (mod 13)

(d) 6 − 7 ≡ x (mod 88)

(e) x ≡ 9 − 19 (mod 7)

5. Find the (representative) solution for x.

(a) 12 + 4 + 7 + 12 + 17 ≡ x (mod 12)

(b) 7 + x ≡ 2 (mod 3)

(c) x ≡ 16 + 2 (mod 8)

(d) x ≡ 57 · 73 (mod 17)

(e) x ≡ 3 · (87 − 5) (mod 7)

(f) x ≡ (−5) − (−11) (mod 10)

6. Compute the following:

(a) 16 + 2 rem 8

(b) 57 · 73 rem 17

(c) 3 · (87 − 5) rem 7

(d) (−5) − (−11) rem 10

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