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# Base 10, Congruence and Residues, and Modular Arithmetic

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1. Two positive integers each have a units digit of 2 when expressed in base 4. Find the units digit of their product when expressed in base 4.

2. Let m be the smallest positive, three-digit integer congruent to 5 (mod 11). Let  be the smallest positive, four-digit integer congruent to 5 (mod 11). What is n-m?

3. Walter, Agnes, and Holly are making beaded lizards. Walter has 476 green beads and 32 red beads. Agnes has 104 green beads and 16 red beads. Holly has 281 green beads and 80 red beads. They all share their beads so as to make the largest possible number of lizards. If a beaded lizard requires 94 green beads and 16 red beads, what is the number of green beads left over?

4. How many integers $$n$$ satisfy $$0 and \(4n\equiv2 (mod 6)$$

5. The marching band has more than 100 members but fewer than 200 members. When they line up in rows of 4 there is one extra person; when they line up in rows of 5 there are two extra people; and when they line up in rows of 7 there are three extra people. How many members are in the marching band?

6. In base b, there are exactly one hundred three-digit numbers whose digits are all distinct. (That's "one hundred" in the ordinary sense, $100_(10)$.)

What is b

7. What is the remainder when $225^66-327^66$ is divided by 17?

Aug 2, 2018
edited by yasbib555  Aug 3, 2018

### 1+0 Answers

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1)

12 in base 4 x 22 in base 4 =330 - the units digit in base 4 =0

2)

9 x 11 + 5 = 104 - the smallest 3-digit number

91 x 11 + 5 =1,006 - the smallest 4-digit number.

5)

M mod 4 = 1

M mod 5 =2

M mod 7 =3, solve for M

M =140 + 17 = 157 members of the marching Band.

7. What is the remainder when $225^66-327^66$ is divided by 17?

The answer =0

Aug 3, 2018