Given number are 1032 and 408
since ,. 1032 is greater than 408

by Euclid's division algorithm ,

1032 = 408×216. (vi)
or , 408=216×1+192. .(v)
or,. 216=192×1+14. (iv)
or,. 192=14×13+10. (iii)
or,. 14=10×1+4. .(ii)
or,. 10=4×2+2. .(i)
or,. 4 =2×2+0

hence HCF of given number is 2

We can write it as,

2= 10-4×2. { from eq(i)}
or. 2=10-(14-10)×2. {from eq (ii)}
or. 2=10-14×2+10×2
or. 2= 10×3-14×2
or. 2=(192-14×13)×3-14×2. { from eq (iii)}
or. 2=192×3-14×16-14×2
or. 2=192×3-14×18
or. 2= 192×3-(216-192)×18. {from eq (iv)}
or. 2=192×3-216×18+192×18
or. 2=192×21-216×18
or. 2=(408-216)×21-216×18. { from eq (v)}
or. 2=408×21-216×21-216×18
or. 2=408×21-216×39
or. 2=408×21-(1032-408×2)×18 {from eq (vi)}
or. 2=408×21-1032×18+408×20
or. 2=408×41-1032×18
or. 2=1032×(-18)+408×41. (A)

since HCF is given in 1032m+408n form
after comparing it with eq(A)
We get,
m =-39. and
n = 41


:-)i wish it will help you

Explanation:

To find the values of m and n in the form 1032m + 408n, we need to calculate the Highest Common Factor (HCF) of 408 and 1032 using the Euclidean algorithm, which states:

If a = bq + r, where a, b, q, and r are integers, then HCF(a, b) = HCF(b, r).

Step 1: Find the HCF of 408 and 1032

Using the Euclidean algorithm:
1032 = 2 * 408 + 216
408 = 1 * 216 + 192
216 = 1 * 192 + 24
192 = 8 * 24 + 0

The last non-zero remainder is 24, so the HCF of 408 and 1032 is 24.

Step 2: Express HCF in the form 1032m + 408n

24 = 216 - 192
24 = 216 - (408 - 216) = 2 * 216 - 408
24 = 2 * (1032 - 408) - 408 = 2 * 1032 - 3 * 408

Therefore, the HCF of 408 and 1032 can be expressed as 24 = 2 * 1032 - 3 * 408.

Step 3: Determine the values of m and n

Comparing the expression with the required form:
m = 2 and n = -3

Therefore, the values of m and n are m = 2 and n = -3, respectively.