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! :)
+113 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
+113 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
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.
+289 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.
