1. What is the smallest integer \(n\), greater than \(1\), such that n^-1 (mod 130) and n^-1 (mod 231) are both defined?

2. For a nonnegative integer \(n\), let \(r_9(n) \) stand for the remainder left when \(n\) is divided by \(9.\) For example, \(r_9(25)=7\)

What is the \(22nd \) entry in an ordered list of all nonnegative integers \(n\) that satisfy

\(r_9(5n)≤4? \) (Note that the first entry in this list is .)

3. What is the greatest three-digit number that is one more than a multiple of 9 and three more than a multiple of 5?

4. 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 \(b\)?

5. Compute 29^13-5^13 modulo 7.

6. 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?

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

8. What is the sum of the units digits of all the multiples of $3$ between $0$ and $50$?

9. What is the smallest positive integer \(n\) such that \(17n\equiv1234 (mod7)? \)

10. For what base-6 digit $d$ is $2dd5_6$ divisible by the base 10 number 11? (Here $2dd5_6$ represents a base-6 number whose first digit is 2, whose last digit is 5, and whose middle two digits are both equal to $d$).

11. If \(k=1/(1+2x)\), where \(x\) is an integer greater than 1 and k can be represented as a terminating decimal, find the sum of all possible values of k.

yasbib555 Aug 10, 2018

#1**-1 **

2)

If I understand your question, then:

5n mod 9 =<4

n =9a + 8, where a=1, 2, 3.........etc.

n=9*1 + 8..............1st pair.

n=9*22 + 8............22nd pair.

3)

The greatest such number is =928

928 - 1 =927 / 9 =103

928 -3 =925 / 5 =185

5. Compute 29^13-5^13 modulo 7.

29^13 - 5^13 =[2.4 x 10^14] mod 7 =2

8)

3 | 6 | 9 | 12 | 15 | 18 | 21 | 24 | 27 | 30 | 33 | 36 | 39 | 42 | 45 | 48

Add up all the units digits to get the sum.

9)

I cannot find an INTEGER solution for n

10)

2445 in base 6 = 605 in base 10, and:

605 / 11 =55

11)

x =2 and k =0.2

x =12 and k =0.04

0.2 + 0.04 =0.24

Guest Aug 10, 2018