If a is congruent to b mod n and k divides n then a is congr

If a is congruent to b mod n and k divides n, then a is congruent to b mod k. Give an example that the converse is false.

Solution

to show converse is false we need to choose a,b,k which satisfies \"a is congruent to b mod k\" then take multiple of k to show that it doesn\'t satisfy \"a is congruent to b mod n\"

take a=29

b=1

k=4

then (29-1)=28 is clearly divisible by 4

now take multiple of k that is 8

so n=8

while 8 doesn\'t divide 28

so it is proved that converse is false :)