Question

if the hcf of 180 and 432 is expressed as (180m+432n),where m and n are intergers,then what is the differnece between m and n

Answer 1

Given :- if the hcf of 180 and 432 is expressed as (180m+432n), where m and n are intergers, then what is the differnece between m and n ?

Solution :-

Prime factors of 180 and 432 is :-

→ 180 = 2 * 2 * 3 * 3 * 5

→ 432 = 2 * 2 * 2 * 2 * 3 * 3 * 3

HCF = 2 * 2 * 3 * 3 = 36.

now , given that, HCF is written as 180m+432n .

So,

→ 180m + 432n = 36 .

→ 36(5m + 12n) = 36

dividing both sides we get,

→ 5m + 12n = 1

Now, we have given that, m and n both are integers.

So,

putting values of m , we get :-

• m = 0, n = (1/12) => n = not a integer.
• m = 1 , n = (-1/3) => n = not a integer.
• m = 2 , n = (-3/4) => n = not a integer.
• m = 3 , n = (-7/6) => n = not a integer.
• m = 4 , n = (-19/12) => n = not a integer.
• m = 5 , n = (-2) => n = An integer.

Hence,

→ Difference between m and n = m - n = 5 - (-2) = 5 + 2 = 7 (Ans.)

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Answer 2

$\underline{\textsf{Given:}}$

$\textsf{ Hcf of 180 and 432 is expressed as (180m+432n)}$

$\underline{\textsf{To find:}}$

$\textsf{Difference between m and n}$

$\underline{\textsf{Solution:}}$

$\textsf{By Euclidean division algorithm, we get}$

$\mathsf{432=2(180)+72}$.......(1)

$\mathsf{180=2(72)+36}$.........(2)

$\mathsf{72=2(36)+0}$

$\textsf{Now (2) can be written as}$

$\mathsf{36=180-2(72)}$

$\textsf{Using (1)}$

$\mathsf{36=180-2[432-2(180)]}$

$\mathsf{36=180-2(432)+4(180)}$

$\mathsf{36=5(180)-2(432)}$

$\textsf{Comparing with 180m+432n we get}$

$\textsf{m=5 and n=-2}$

$\textsf{Then, m-n=5-(-2)=5+2=7}$

$\underline{\textsf{Answer:}}$

$\textsf{The differeence between m and n is 7}$