Question

what is Fundamental theoram of Anithmetic​

Answer 1

ғᴜɴᴅᴀᴍᴇɴᴛᴀʟ ᴛʜᴇᴏʀᴇᴍ ᴏғ ᴀʀɪᴛʜᴍᴇᴛɪᴄ sᴛᴀᴛᴇs ᴛʜᴀᴛ ᴇᴠᴇʀʏ ɪɴᴛᴇɢᴇʀ ɢʀᴇᴀᴛᴇʀ ᴛʜᴀɴ 1 ɪs ᴇɪᴛʜᴇʀ ᴀ ᴘʀɪᴍᴇ ɴᴜᴍʙᴇʀ ᴏʀ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇssᴇᴅ ɪɴ ᴛʜᴇ ғᴏʀᴍ ᴏғ ᴘʀɪᴍᴇs. ɪɴ ᴏᴛʜᴇʀ ᴡᴏʀᴅs, ᴀʟʟ ᴛʜᴇ ɴᴀᴛᴜʀᴀʟ ɴᴜᴍʙᴇʀs ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇssᴇᴅ ɪɴ ᴛʜᴇ ғᴏʀᴍ ᴏғ ᴛʜᴇ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ɪᴛs ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀs. ᴛᴏ ʀᴇᴄᴀʟʟ, ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀs ᴀʀᴇ ᴛʜᴇ ɴᴜᴍʙᴇʀs ᴡʜɪᴄʜ ᴀʀᴇ ᴅɪᴠɪsɪʙʟᴇ ʙʏ 1 ᴀɴᴅ ɪᴛsᴇʟғ ᴏɴʟʏ. ғᴏʀ ᴇxᴀᴍᴘʟᴇ, ᴛʜᴇ ɴᴜᴍʙᴇʀ 35 ᴄᴀɴ ʙᴇ ᴡʀɪᴛᴛᴇɴ ɪɴ ᴛʜᴇ ғᴏʀᴍ ᴏғ ɪᴛs ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀs ᴀs:

35 = 7 × 5

ʜᴇʀᴇ, 7 ᴀɴᴅ 5 ᴀʀᴇ ᴛʜᴇ ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀs ᴏғ 35

sɪᴍɪʟᴀʀʟʏ, ᴀɴᴏᴛʜᴇʀ ɴᴜᴍʙᴇʀ 114560 ᴄᴀɴ ʙᴇ ʀᴇᴘʀᴇsᴇɴᴛᴇᴅ ᴀs ᴛʜᴇ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ɪᴛs ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀs ʙʏ ᴜsɪɴɢ ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴍᴇᴛʜᴏᴅ,

114560 = 27 × 5 × 179

sᴏ, ᴡᴇ ʜᴀᴠᴇ ғᴀᴄᴛᴏʀɪᴢᴇᴅ 114560 ᴀs ᴛʜᴇ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴛʜᴇ ᴘᴏᴡᴇʀ ᴏғ ɪᴛs ᴘʀɪᴍᴇs.

ᴛʜᴇʀᴇғᴏʀᴇ, ᴇᴠᴇʀʏ ɴᴀᴛᴜʀᴀʟ ɴᴜᴍʙᴇʀ ᴄᴀɴ ʙᴇ ᴇxᴘʀᴇssᴇᴅ ɪɴ ᴛʜᴇ ғᴏʀᴍ ᴏғ ᴛʜᴇ ᴘʀᴏᴅᴜᴄᴛ ᴏғ ᴛʜᴇ ᴘᴏᴡᴇʀ ᴏғ ɪᴛs ᴘʀɪᴍᴇs. ᴛʜɪs sᴛᴀᴛᴇᴍᴇɴᴛ ɪs ᴋɴᴏᴡɴ ᴀs ᴛʜᴇ ғᴜɴᴅᴀᴍᴇɴᴛᴀʟ ᴛʜᴇᴏʀᴇᴍ ᴏғ ᴀʀɪᴛʜᴍᴇᴛɪᴄ, ᴜɴɪϙᴜᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴛʜᴇᴏʀᴇᴍ ᴏʀ ᴛʜᴇ ᴜɴɪϙᴜᴇ-ᴘʀɪᴍᴇ-ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴛʜᴇᴏʀᴇᴍ.

ɪ ʜᴏᴘᴇ ɪᴛ ᴄᴀɴ ʜᴇʟᴘ ʏᴏᴜ

✦✧✧ ʜᴀᴠᴇ ᴀ ʙᴇᴀᴜᴛɪғᴜʟ ᴅᴀʏ ᴀʜᴇᴀᴅ ✧✧✦

Answer 2

The fundamental theorem of arithmetic states that every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes (Hardy and Wright 1979, pp. This theorem is also called the unique factorization theorem