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1) How many positive three-digit integers with a  5 in the units place are divisible by 15 ?

 

2) If a and b are integers such that \(ab\equiv 17\pmod{20}\), then what is the remainder when (a+10)(b+10) is divided by 20?

 

THANKS! :)

 Aug 6, 2022
 #1
avatar+113 
+1

On ex. 1 python says:

There are 33 integers in that group, they are:
15, 45, 75, 105, 135, 165, 195, 225, 255, 285, 315, 345,
375, 405, 435, 465, 495, 525, 555, 585, 615, 645, 675,
705, 735, 765, 795, 825, 855, 885, 915, 945, 975

 

See https://www.online-python.com/nxgpkMajEo

 

On ex. 2:

Turning to mod,

in python (and other programming languages)

% is used as the modulus operator

 

ab%20 = 17

(a + 10) x (b + 10)%20 =

(ab +10a + 10b + 100)%20 =

ab%20 + 10a%20 + 10b%20 + 100%20 =

17 + 10 + 10 + 0 = 17 + 20 = 37 (assuming a and b not divisible by 20)

But 37%20 = 17

 Aug 6, 2022
edited by tuffla2022  Aug 6, 2022
edited by tuffla2022  Aug 6, 2022
edited by tuffla2022  Aug 6, 2022
edited by tuffla2022  Aug 6, 2022
 #3
avatar+113 
+1

Sorry, didn't notice that there had to be 3 digits. I stand corrected by Guest below. Code has been updated.

 

Exercise 1:
There are 30 integers in that group, they are:
105, 135, 165, 195, 225, 255, 285, 315, 345, 375, 405, 435,
465, 495, 525, 555, 585, 615, 645, 675, 705, 735, 765,
795, 825, 855, 885, 915, 945, 975

See https://www.online-python.com/YbUHkOvQyx

tuffla2022  Aug 6, 2022
 #2
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+1

1) - There are:

 

(105, 135, 165, 195, 225, 255, 285, 315, 345, 375, 405, 435, 465, 495, 525, 555, 585, 615, 645, 675, 705, 735, 765, 795, 825, 855, 885, 915, 945, 975) >>Total = 30 such integers.

 Aug 6, 2022
 #4
avatar+251 
+1

Using python codes is a great way to efficiently solve them but we are focused on problem solving our own minds here.

 

1. 15 = 5 *3

Since the three digit numbers have a unit digits of 5, we just need the number in the form of AB5, to be divisible by 3. And in order for a number to be divisible by 3, its digits need to add up to a multiple of 3. 

 

A + B + 5 = 0 (mod 3)

A + B = 1 (mod 3)

 

Can you solve the question from here and find all digits A and B that make the congruence true? (Hint, casework!)

 

2. One simple way to solve this is by using the loss of generality, plug in any numbers a and b, where their product leaves a remainder of 17 when divided by 20. After finding those numbers, plug it into (a + 10)(b + 10) and see the remainder.

 Aug 6, 2022

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