What is the smallest odd natural number that leaves aremainder of 2 when divided

Question

What is the smallest odd natural number that leaves aremainder of 2 when divided by 3 and a remainder of 3 when dividedby 5? I know it needs to be set up x=2(mod 3) x=3(mod5) but somehow they say 8 is a solution and the smallest positiveodd solution is 23. can someone please walk me through how to do this? What is the smallest odd natural number that leaves aremainder of 2 when divided by 3 and a remainder of 3 when dividedby 5? I know it needs to be set up x=2(mod 3) x=3(mod5) but somehow they say 8 is a solution and the smallest positiveodd solution is 23. can someone please walk me through how to do this?

Explanation / Answer

QuestionDetails: What is the smallest odd natural number that leaves aremainder of 2 when divided by 3 and a remainder of 3 when dividedby 5? I know it needs to be set up x=2(mod 3)............................1 x=3(mod5)....................................2
OK
LET US FIND A GENERAL SOLUTION AND THEN THE SMALLEST ODD NATURALNUMBER
FROM EQN.1 WE GET
X=2+3P .................3
WHERE P IS AN INTEGER. PUTTING IN EQN.2 WE GET
2+3P=3[MOD 5]
3P=1[MOD 5]
0 =5 [MOD 5]
ADDING
3P=6[MOD 5]
P= 2[MOD 5]
P=2+5Q.....................4
WHERE Q IS AN INTEGER .PUTTING IN EQN.3
X=2+3[2+5Q]= 8+15Q
SO THE SOLUTION SET IS OBTAINED BY PUTTING Q=0,1,2....ETC...
THAT IS
X=[8,23,38,53.....ETC]
SO IF WE WANT THE SMALLEST NATURAL NUMBER IT IS X=8 AS YOUSAID.
BUT IF WE WANT SMALLEST ODD NATURAL NUMBER, AS ASKED FOR IN THE
PROBLEM, THEN THE ANSWER IS X=23
but somehow they say 8 is a solution and the smallest positive oddsolution is 23. can someone please walk me through how to do this?.......HOPEYOU GOT IT NOW ?
IF STILL IN DOUBT PLEASE COME BACK

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