Question

WILL GIVE BRAINLIEST!!!

Answer 1

Answer:

6 total

3 positive 2 digit numbers: 17, 52, 87

3 negative 2 digit numbers: -18, -53,-88

Step-by-step explanation:

We are given

x ≡ 3(mod7), x ≡ 2(mod5) and we want x to be two digits long.

x ≡ 3(mod7) means x-3=7k and

x ≡ 2(mod5) means x-2=5m where k and m are integers.

We need to find an x that satisfies both.

So some multiples of 7 are 7,14,21,28,35,42,49,56,63,70,77,84,91,98

Some multiples or 5 are

5,10 15,20,25,30 35,40,45,45,50,55,60,65,70

Now add 3 to first list for x=7k+3

10,17,24,31,38,45,52,59,66,73,80,87,94,101

Now add 2 to second list for x=5m+2

7,12 17,22,27,32 37,42,47,52,57,62,67,72

We only want to look at 2 digit numbers... just need to expand second list more:

Now add 2 to second list for x=5m+2

7,12,17,22,27,32,37,42,47,52,57,62,67,72,77,82,87,92,97,

That's all we need.

So let's write our lists and see the common numbers in them:

List 1: 10,17,24,31,38,45,52,59,66,73,80,87,94,101

List 2:

7,12,17,22,27,32,37,42,47,52,57,62,67,72,77,82,87,92,97

Common numbers: 17, 52, 87

Considering negative numbers as well:

x=7k+3:

3, -4, -11, -18, -25, -32,-39, -46, -53, -60, -67, -74, -81, -88, -95

x=5m+2:

2,-3,-8,-13,-18,-23,-28,-33,-38,-43,-48,-53,-58,-63,-68,-73,-78,-83,-88,-93,-98

Common 2digit numbers:

-18, -53,-88