Question

Using Euclid algorithm lemma to find square of any integer 5p 5p+1

Answer 1

Answer:

ᴀɴꜱᴡᴇʀ. ʟᴇᴛ x ʙᴇ ᴀɴʏ ᴘᴏꜱɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀ ᴛʜᴇɴ x = 5Q ᴏʀ x = 5Q+1 ᴏʀ x = 5Q+4 ꜰᴏʀ ɪɴᴛᴇɢᴇʀ x. ɪꜰ x = 5Q, x2 = (5Q)2 = 25Q2 = 5(5Q2) = 5ɴ (ᴡʜᴇʀᴇ ɴ = 5Q2 ) ɪꜰ x = 5Q+1, x2 = (5Q+1)2 = 25Q2+10Q+1 = 5(5Q2+2Q)+1 = 5ɴ+1 (ᴡʜᴇʀᴇ ɴ = 5Q2+2Q ) ɪꜰ x = 5Q+4, x2 = (5Q+4)2 = 25Q2+40Q+16 = 5(5Q2 + 8Q + 3)+ 1 = 5ɴ+1 (ᴡʜᴇʀᴇ ɴ = 5Q2+8Q+3 ) ∴ɪɴ ᴇᴀᴄʜ ᴏꜰ ᴛʜʀᴇᴇ ᴄᴀꜱᴇꜱ x2 ɪꜱ ᴇɪᴛʜᴇʀ ᴏꜰ ᴛʜᴇ ꜰᴏʀᴍ 5Q ᴏʀ 5Q+1 ᴏʀ 5Q+4 ᴀɴᴅ ꜰᴏʀ ɪɴᴛᴇɢᴇʀ Q.

Step-by-step explanation:

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