Question

find the hcf of 65 and 117 and express it in the form of 65m +117n.
find the largest no that divide 246 and 1029 and leave the remainder as 5.
GIVE ME THE ANSWER OF THESE ALL QUESTIONS WITH PROOF AND IF IT IS IN A STANDARD WAY I WILL GIVE THE OWNER OF THE ANSER AS A BRAINLIST ONE.
BEST OF LUCK

Answer 1

(1) Given Integers are 65 and 117.

Since 117 > 65, we apply Division Lemma to 117 and 65 to get,

117 = 65 * 1 + 52

Again applying division lemma to 65 and 52, we get

65 = 52 * 1 + 13

Once again applying division lemma to 52 and 13, we get

52 = 4 * 13 + 0

Since remainder has now become 0, this implies that HCF of 65 and 117 = 13.

Therefore the HCF of 65 and 117 is 13.

Now, 13 = 65 - 52 * 1   -------- (1)

          52 = 117 - 65 * 1  --------- (2)

          Substitute (2) in (1), we get

         13 = 65 - (117 - 65 * 1) * 1

              = 65 - 117 + 65 * 1

              = 65 * 2 + 117 * (-1)


Therefore HCF of 65 and 117 in the form of 65m + 117n is m = 2 and n = -1.



(2)  Given that required number, when divides 245 and 1029, the remainder is 5 in each case.

We need to subtract numbers from the remainder.

245 - 5 = 240

1029 - 5 = 1024. 


= > 240 = 2 * 2 * 2 * 2 * 3 * 5

= > 1024 = 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2 * 2

Therefore the HCF of 241 and 1024 = 2 * 2 * 2 * 2

                                                            = 16.


Therefore the largest number which divides 245 and 1029 leaving remainder 5 in each case is 16.


Hope this helps!