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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.

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.

Aug 10, 2018
edited by yasbib555  Aug 10, 2018

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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

Aug 10, 2018