Question

use euclid's division lemma to show that the cube of any positive integer is either of the form 9m,9m + 1 ,or 9m + 8​

Answer 1

Step-by-step explanation:

⭐Let a ɪs ᴀɴʏ ᴘᴏsɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀ ᴀɴᴅ ʙ = 3

⭐ ᴛʜᴇɴ , ʙʏ ᴇᴜᴄʟɪᴅ's ᴅɪᴠɪsɪᴏɴ ʟᴇᴍᴍᴀ , ᴡᴇ

ʜᴀᴠᴇ ᴀ = 3ϙ + ʀ , ғᴏʀ sᴏᴍᴇ ɪɴᴛᴇɢᴇʀ ϙ ≥ 0

ᴀɴᴅ 0 ≤ ʀ < 3.

[ᴛᴇx]➩ {ᴀ}^{3 } = (3ϙ + ʀ) ^{3} ᴡʜᴇʀᴇ \: 0≤ʀ < 3[/ᴛᴇx]

[ᴛᴇx]➩ {ᴀ}^{3} =( 27 {ϙ}^{3} + 27 {ϙ}^{2} ʀ + 9ϙ {ʀ}^{2} ) + {ʀ}^{3} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: .......(1)[/ᴛᴇx]

ஃ ( ᴀ + ʙ ) ^ 3 = ᴀ^3 + ʙ^3 + 3ᴀʙ( ᴀ + ʙ)

⭐ ɴᴏᴡ, ᴘᴏssɪʙʟᴇ ᴠᴀʟᴜᴇs ᴏғ ʀ ᴀʀᴇ 0 , 1 & 2.

[ᴛᴇx]\ᴘɪɴᴋ{ᴄᴀsᴇ \: 1\::} \: ᴡʜᴇɴ \: ʀ \: = 0 [/ᴛᴇx]

⭐ ᴘᴜᴛᴛɪɴɢ ʀ = 0 ɪɴ ( 1 ) , ᴡᴇ ɢᴇᴛ

[ᴛᴇx]➩ {ᴀ}^{3 } = 27 {ϙ}^{3} = 9 ( 3ϙ ^{3} ) = 9ᴍ \\ \\ ᴡʜᴇʀᴇ \: \: \: ᴍ = 3 {ϙ}^{3} ɪs \: ᴀɴ \: ɪɴᴛᴇɢᴇʀ[/ᴛᴇx]

[ᴛᴇx]\ᴘɪɴᴋ{ᴄᴀsᴇ \: 2\::} \: ᴡʜᴇɴ \: ʀ \: = 1 [/ᴛᴇx]

⭐ ᴘᴜᴛᴛɪɴɢ ʀ = 1 ɪɴ ( 1 ) , ᴡᴇ ɢᴇᴛ

[ᴛᴇx]➩ {ᴀ}^{3} = 9ᴍ + 1 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ ᴡʜᴇʀᴇ \: ᴍ = ϙ(3 {ϙ }^{2} + 6ϙ + 1) ɪs \: ᴀɴ \: ɪɴᴛᴇɢᴇʀ[/ᴛᴇx]

[ᴛᴇx]\ᴘɪɴᴋ{ᴄᴀsᴇ \: 3\::} \: ᴡʜᴇɴ \: ʀ \: = 2[/ᴛᴇx]

⭐ ᴘᴜᴛᴛɪɴɢ ʀ = 2 ɪɴ ( 1 ) , ᴡᴇ ɢᴇᴛ

[ᴛᴇx]➩ {ᴀ}^{3} = 9ᴍ +8 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ ᴡʜᴇʀᴇ \: ᴍ = ϙ(3 {ϙ }^{2

Answer 2

Answer:

Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m,9m+1 or 9m+8. Answer: Let us consider a and b where a be any positive number and b is equal to 3. so r is an integer greater than or equal to 0 and less than 3

Step-by-step explanation:

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