Question

. The HCF of 90a²xy and 81 xy

Answer 1

Answer:

ɢɪᴠᴇɴ ɪɴᴛᴇɢᴇʀs ᴀʀᴇ 81 ᴀɴᴅ 237 sᴜᴄʜ ᴛʜᴀᴛ 81<237.

ᴀᴘᴘʟʏɪɴɢ ᴅɪᴠɪsɪᴏɴ ʟᴇᴍᴍᴀ ᴛᴏ 81 ᴀɴᴅ 237, ᴡᴇ ɢᴇᴛ

237=81×2+75 

sɪɴᴄᴇ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ 75=0. sᴏ, ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ ᴅɪᴠɪsᴏʀ 81 ᴀɴᴅ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ 75 ᴀʀɴᴅᴀᴘᴘʟʏ ᴅɪᴠɪsɪᴏɴ ʟᴇᴍᴍᴀ ᴛᴏ ɢᴇᴛ

81=75×1+6  

ᴡᴇ ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ ɴᴇᴡ ᴅɪᴠɪsᴏʀ 75 ᴀɴᴅ ᴛʜᴇ ɴᴇᴡ ʀᴇᴍᴀɪɴᴅᴇʀ 6 ᴀɴᴅ ᴀᴘᴘʟʏ ᴅɪᴠɪsɪᴏɴ ʟᴇᴍᴍᴀ ᴛᴏ ɢᴇᴛ

75=6×12+3  

ᴡᴇ ᴄᴏɴsɪᴅᴇʀ ᴛʜᴇ ɴᴇᴡ ᴅɪᴠɪsᴏʀ 6 ᴀɴᴅ ᴛʜᴇ ɴᴇᴡ ʀᴇᴍᴀɪɴᴅᴇʀ 3 ᴀɴᴅ ᴀᴘᴘʟʏ ᴅɪᴠɪsɪᴏɴ ʟᴇᴍᴍᴀ ᴛᴏ ɢᴇᴛ

6=3×2+0  

ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ ᴀᴛ ᴛʜɪs sᴛᴀɢᴇ ɪs ᴢᴇʀᴏ. sᴏ, ᴛʜᴇ ᴅɪᴠɪsᴏʀ ᴀᴛ ᴛʜɪs sᴛᴀɢᴇ ᴏʀ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ ᴀᴛ ᴛʜᴇ ᴇᴀʀʟɪᴇʀ sᴛᴀɢᴇ ɪ.ᴇ. 3 ɪs ᴛʜᴇ ʜᴄғ ᴏғ 81 ᴀɴᴅ 237.

ᴛᴏ ʀᴇᴘʀᴇsᴇɴᴛ ᴛʜᴇ ʜᴄғ ᴀs ᴀ ʟɪɴᴇᴀʀ ᴄᴏᴍʙɪɴᴀᴛɪᴏɴ ᴏғ ᴛʜᴇ ɢɪᴠᴇɴ ᴛᴡᴏ ɴᴜᴍʙᴇʀs, ᴡᴇ sᴛᴀʀᴛ ғʀᴏᴍ ᴛʜᴇ ʟᴀsᴛ ʙᴜᴛ ᴏɴᴇ sᴛᴇᴘ ᴀɴᴅ sᴜᴄᴄᴇssɪᴠᴇʟʏ ᴇʟɪᴍɪɴᴀᴛᴇ ᴛʜᴇ ᴘʀᴇᴠɪᴏᴜs ʀᴇᴍᴀɪɴᴅᴇʀ ᴀs ғᴏʟʟᴏᴡs: 

ғʀᴏᴍ (ɪɪɪ), ᴡᴇ ʜᴀᴠᴇ

3=75−6×12

⇒3=75−(81−75×1)×12

⇒3=75−12×81+12×75

⇒3=13×75−12×81

⇒3=13×(237−81×2)−12×81

⇒3=13×237−26×81−12×81

⇒3=13×237−38×81

⇒3=237x+81ʏ, ᴡʜᴇʀᴇ x=13 ᴀɴᴅ ʏ=−38.

ɴᴏᴡ ᴛʜᴇ ʜᴄғ (sᴀʏ ᴅ) ᴏғ ᴛᴡᴏ ᴘᴏsɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀs ᴀᴀɴᴅ ʙ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇssᴇᴅ ᴀs ᴀ ʟɪɴᴇᴀʀ ᴄᴏᴍʙɪɴᴀᴛɪᴏɴ ᴏғ ᴀ ᴀɴᴅ ʙ ɪ.ᴇ., ᴅ=xᴀ+ʏʙ ғᴏʀ sᴏᴍᴇ ɪɴᴛᴇɢᴇʀs x ᴀɴᴅ ʏ. 

ᴀʟsᴏ, ᴛʜɪs ʀᴇᴘʀᴇsᴇɴᴛᴀᴛɪᴏɴ ɪs ɴᴏᴛ ᴜɴɪǫᴜᴇ. ʙᴇᴄᴀᴜsᴇ,

ᴅ=xᴀ+ʏʙ

⇒ᴅ=xᴀ+ʏʙ+ᴀʙ−ᴀʙ

⇒ᴅ=(x+ʙ)ᴀ+(ʏ−ᴀ)ʙ

ɪɴ ᴛʜᴇ ᴀʙᴏᴠᴇ ᴇxᴀᴍᴘʟᴇ, ᴡᴇ ʜᴀᴅ

3=13×237−38×81

⇒3=13×237−38×81+237×81−237×81

⇒3=(13×237+237×81)+(−38×81−237×81)

⇒3=(13+81)×237+(−38−237)×81

⇒3=94×237−275×81

⇒3=94×237+(−275)×81