Question

If the hcf of 180 and 432 is expressed as (180m+432n),where m and n are integers, then what is the difference between m and n​

Answer 1

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

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

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

$\textsf{Difference between m and n}$

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

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

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

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

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

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

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

$\textsf{Using (1)}$

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

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

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

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

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

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

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

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

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Answer 2

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 :

→ 2180 = 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 by 36 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 andn = m - n = 5-(-2)= 5+2=7 (Ans.)

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