Problem 19. What is the greatest possible value of the number N that would make the following
statement true: "N numbers can always be found among 97 random integers, in a way that the
difference of each pair would be divisible by 8.”?
statement true: "N numbers can always be found among 97 random integers, in a way that the
difference of each pair would be divisible by 8.”?
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Solution:
N ɴᴜᴍʙᴇʀ ᴄᴀɴ ᴀʟᴡᴀʏs ʙᴇ ғᴏᴜɴᴅ ᴀᴍᴏɴɢ 97 ʀᴀɴᴅᴏᴍ ɪɴᴛᴇɢᴇʀs, ɪɴ ᴀ ᴡᴀʏ ᴛʜᴀᴛ ᴛʜᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ᴏғ ᴇᴀᴄʜ ᴘᴀɪʀ ᴡᴏᴜʟᴅ ʙᴇ ᴅɪᴠɪsɪʙʟᴇ ʙʏ 8.
Lᴇᴛ ᴛʜᴇ ɪɴᴛᴇɢᴇʀs ʙᴇ x, x+8, x+16...
Assume x is not divisible by 8
Sᴏ, ᴛʜᴇʀᴇ ᴀʀᴇ 96 ᴛᴇʀᴍs ᴏᴜᴛ ᴏғ 97 ᴛʜᴀᴛ sᴀᴛɪsғʏ ᴛʜɪs (because x is not divisible by 8) Tʜᴇʀᴇғᴏʀᴇ, ʜɪɢʜᴇsᴛ ᴠᴀʟᴜᴇ ᴡɪʟʟ ʙᴇ:
=> 96/8 + 1 [Wᴇ ʜᴀᴠᴇ ᴇxᴄʟᴜᴅᴇᴅ "x" sᴏ ᴡᴇ ɴᴇᴇᴅ ᴛᴏ ᴀᴅᴅ 1".]
=> 12 + 1
=> 13.
Fᴇᴇʟ ғʀᴇᴇ ᴛᴏ ᴀsᴋ ᴅᴏᴜʙᴛs.
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