Show that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q some integer. ​

Question

Show that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q
some integer.

in progress 0
Audrey 1 month 2021-08-20T19:57:58+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-08-20T19:59:20+00:00

    \underline\mathfrak\purple{By\:Euclid\:Division\:Lemma\::-}

    if ‘a’ and ‘b’ are two positive integers then

    \fcolorbox{red}{yellow}{a\:=\:bq+r}

    where,

    0 ≤ r < b

    let a be a positive integers and b = 6

    \fcolorbox{red}{yellow}{a\:=\:6q+r}

    where,

    0 ≤ r < 6

    from above we can say that remainder is less than 6 and equal or greater than 0

    so the possible values of r be

    \mathfrak\green{0,1,2,3,4,5}

    by putting values of divisor(b) and remainder in equation \fbox\color{red}{a\:=\:bq+r}

    putting r = 0, a = 6q + 0

    a = 6q

    it is divided by 2 so it is an \underline\mathfrak\blue{even} number.

    putting r = 1, a = 6q + 1

    6q is divided by 2 but 1 is not divided.

    it is not divided by 2 so it is an \underline\mathfrak\blue{odd} number.

    putting r = 2, a = 6q + 2

    it is divided by 2 so it is an \underline\mathfrak\blue{even} number.

    putting r = 3, a = 6q + 3

    it is not divided by 2 so it is an \underline\mathfrak\blue{odd} number.

    putting r = 4, a = 6q + 4

    it is divided by 2 so it is an \underline\mathfrak\blue{even} number

    putting r = 5, a = 6q + 5

    it is not divided by 2 so it is an \underline\mathfrak\blue{odd} number.

    so by these we can say that any positive odd integer is of the form 6q+1, or 69 +3, or 69 + 5, where q is some integer.

     <marquee behaviour = left> please mark as brainliest.......

    0
    2021-08-20T19:59:32+00:00

    lets us start with taking a,where a is a postive off integer

    We apply the division algorithm with a and b=6

    since 0 less than or equal to r or lessthan 6 the possible remainders are 012345

    that is

    a can be

    6q+1or6q+3or6q+5

    where quotient

    however since a is odd a cannot be 6q,6q+2,6q+4since they are divisble by 2

    therefore the any odd integer is of the form of 6q+1,6q+3,6q+5

    Thank You

Leave an answer

Browse
Browse

18:9+8+9*3-7:3-1*13 = ? ( )