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
