proof by \" If and only If \" the following: Please, use only the special kind o

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proof by " If and only If " the following: Please, use only the special kind of theorem " IF ANDONLY IF: An integer is a multiple of 5 if and only if its unit digit iseither a 5 or a 0. Please, can someone write this proof indetail? I need everydetail of it. I need to understandit. proof by " If and only If " the following: Please, use only the special kind of theorem " IF ANDONLY IF: An integer is a multiple of 5 if and only if its unit digit iseither a 5 or a 0. Please, can someone write this proof indetail? I need everydetail of it. I need to understandit.

Explanation / Answer

QuestionDetails: proof by " If and only If " the following: Please, can someone write this proof indetail? I need everydetail of it. I need to understand it.
PROPOSITION
AN INTEGER IS A MULTIPLE OF 5...
TST....IT HAS 0 OR 5 IN UNITS PLACE.
PROOF...
ANY INTEGER ENDS IN
0,1,2,3,4,5,6,7,8,9 IN UNITS PLACE
WHEN MULTIPLIED WITH 5 THEN UNITS PLACE WOULD BE
5*0=0.........THAT IS ...................................0 IN UNITSPLACE
OR
5*1=5.........THAT IS ...................................5 IN UNITSPLACE
OR
5*2=10.........THAT IS ...................................0 INUNITS PLACE
OR
5*3=15.........THAT IS ...................................5 INUNITS PLACE
OR
5*4=20.........THAT IS ...................................0 INUNITS PLACE
OR
5*5=25.........THAT IS ...................................5 INUNITS PLACE
OR
5*6=30.........THAT IS ...................................0 INUNITS PLACE
OR
5*7=35.........THAT IS ...................................5 INUNITS PLACE
OR
5*8=40.........THAT IS ...................................0 INUNITS PLACE
OR
5*9=45.........THAT IS ...................................5 INUNITS PLACE
HENCE IN ALL CASES UNITS PLACE IS EITHER ZERO OR 5.....PROVED
CONVERSE
AN INTEGER HAS 0 OR 5 IN UNITS PLACE
TO SHOW THAT IT IS A MULTIPLE OF 5
CASE 1 ....
0 IN UNITS PLACE
LET THE INTEGER BE
X0=X*10+0=10*X...................., WHERE X COULD BE ANYINTEGER...
SINCE 10 IS DIVISIBLE BY 5 , OBVIOUSLY X0=X*10 IS DIVISIBLE BY5....PROVED
CASE 2 .....
5 IN UNIT PLACE
X5=X*10+5=10X+5...................., WHERE X COULD BE ANYINTEGER..
WE KNOW THAT WHEN X IS DIVIDED BY 5 WE COULD GET 0,1,2,3 AND 4 ASREMINDER
CONSIDER ALL POSSIBILITIES .
X=0+5P=5P.....WHERE P IS AN INTEGER......THENX5=10X+5=10*5P+5=50P+5=5[10P+1]=5*INTEGER
HENCE IT IS DIVISIBLE BY 5 ....OK
X=1+5P.....WHERE P IS AN INTEGER......THENX5=10X+5=10*5P+10+5=50P+15=5[10P+3]=5*INTEGER
HENCE IT IS DIVISIBLE BY 5 ....OK
X=2+5P.....WHERE P IS AN INTEGER......THENX5=10X+5=10*5P+10*2+5=50P+25=5[10P+5]=5*INTEGER
HENCE IT IS DIVISIBLE BY 5 ....OK
X=3+5P.....WHERE P IS AN INTEGER......THENX5=10X+10*3+5=10*5P+35=50P+5=5[10P+7]=5*INTEGER
HENCE IT IS DIVISIBLE BY 5 ....OK
X=4+5P.....WHERE P IS AN INTEGER......THENX5=10X+5=10*5P+10*4+5=50P+45=5[10P+9]=5*INTEGER
HENCE IT IS DIVISIBLE BY 5 ....OK
THUS IN ALL,CASES, THE NUMBER IS DIVISIBLE BY 5.
PROVED

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