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$\sqrt{5is \: irrerational}$
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Answer 1

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Answer 2

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$\huge\mathbb\blue{\underline{SOLUTION}}$

➪ ʟᴇᴛs ᴀssᴜᴍᴇ , ᴛᴏ ᴛʜᴇ ᴄᴏɴᴛʀᴀʀʏ , ᴛʜᴀᴛ √5 ɪs ʀᴀᴛɪᴏɴᴀʟ .

ɪ.ᴇ., ᴀ & ʙ ( ᴇǫᴜᴀʟ ɴᴏᴛ ᴛᴏ 0 ) sᴜᴄʜ ᴛʜᴀᴛ √5 = ᴀ/ʙ

☞︎︎︎ sᴜᴘᴘᴏsᴇ , ᴀ & ʙ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ ᴏᴛʜᴇʀ ᴛʜᴀɴ 1 , ᴛʜᴇɴɪᴛ ᴄᴀɴ ʙᴇ ᴅɪᴠɪᴅᴇᴅ ʙʏ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ , ᴀssᴜᴍᴇ ᴛʜᴀᴛ ᴀ & ʙ ᴀʀᴇ ᴄᴏᴘʀɪᴍᴇ .

sᴏ , ʙ√5 = ᴀ

• sǫᴜᴀʀɪɴɢ ᴏɴ ʙᴏᴛʜ sɪᴅᴇs , ᴡᴇ ɢᴇᴛ

5ʙ^2 = ᴀ^2

» Tʜᴇʀᴇғᴏʀᴇ , ᴀ^2 ɪs ᴅɪᴠɪsɪʙʟᴇ ʙʏ 5 .

ᴀ ɪs ᴀʟsᴏ ᴅɪᴠɪsɪʙʟᴇ ʙʏ 5 .

~ sᴏ , ᴡᴇ ᴄᴀɴ ᴡʀɪᴛᴇ ᴀ = 5ᴄ ғᴏʀ sᴏᴍᴇ ɪɴᴛᴇɢᴇʀ ᴄ .

° sᴜʙsᴛɪᴛᴜᴛɪɴɢ ғᴏʀ ᴀ , ᴡᴇ ɢᴇᴛ 5ʙ^2 = 25ᴄ^2

➡️ ᴛʜɪs ᴍᴇᴀɴs ᴛʜᴀᴛ ʙ^2 ɪs ᴅɪᴠɪsɪʙʟᴇ ʙʏ 5 , ᴀɴᴅ sᴏ ʙ ɪs ᴀʟsᴏ ᴅɪᴠɪsɪʙʟᴇ ʙʏ 3 .

➡️ ᴛʜᴇʀᴇғᴏʀᴇ , ᴀ & ʙ ʜᴀᴠᴇ ᴀᴛ ʟᴇᴀsᴛ 5 ᴀs ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ .

⏩ ʙᴜᴛ ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴛʜᴇ ғᴀᴄᴛ ᴛʜᴀᴛ ᴀ& ʙ ᴀʀᴇ ᴄᴏᴘʀɪᴍᴇ .

• ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ ʜᴀs ᴀʀɪsᴇɴ ʙᴇᴄᴀᴜsᴇ ᴏғ ɪɴᴄᴏʀʀᴇᴄᴛ ᴀssᴜᴍᴘᴛɪᴏɴ ᴛʜᴀᴛ √5 ɪs ʀᴀᴛɪᴏɴᴀʟ .

✓ sᴏ , ᴡᴇ ᴄᴏɴᴄʟᴜᴅᴇ ᴛʜᴀᴛ √5 ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ .