Question

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Answer 1

(4) 

prime factorization of 4 = 2 * 2

Prime factorization of 7 = 7

Prime factorization of 13 = 13

LCM(4,7,13) = 2 * 2 * 7 * 13

                    = 364.



We know that the greatest 4-digit number is 9999.

On dividing 9999 by 364, 

364)9999(27

       728

       --------

        2719

        2548

       ---------

            171


Therefore, the greatest 4-digit number divisible by 4,7 and 13 = 9999 - 171

                                                                                                           = 9828


Therefore the largest 4-digit number leaves a remainder 3 = 9828 + 3

                                                                                                     = 9831.




(5) 

First, we need to find the LCM of 5,6,4 and 3.

Prime factorization of 5 = 5

Prime factorization of 6 = 2 * 3

Prime factorization of  4 = 2 * 2

Prime factorization of 3 = 3

LCM(5,6,4,3) = 2 * 2 * 3 * 5

                      = 60.


In order to find the remainder, we should divide 2497 with the LCM.

60) 2497( 41

      240

      ---------
 
            97

            60

      -----------

               37.



Therefore the least number which should be added = (60 - 37)

                                                                                        = 23.



Hope this helps!

Answer 2

(4) 

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ 4 = 2 * 2

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ 7 = 7

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ 13 = 13

ʟᴄᴍ(4,7,13) = 2 * 2 * 7 * 13

                    = 364.

ᴡᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ ᴛʜᴇ ɢʀᴇᴀᴛᴇsᴛ 4-ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ɪs 9999.

ᴏɴ ᴅɪᴠɪᴅɪɴɢ 9999 ʙʏ 364, 

364)9999(27

       728

       --------

        2719

        2548

      ---------

            171

ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜᴇ ɢʀᴇᴀᴛᴇsᴛ 4-ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ᴅɪᴠɪsɪʙʟᴇ ʙʏ 4,7 ᴀɴᴅ 13 = 9999 - 171

                                                                                                           = 9828

ᴛʜᴇʀᴇғᴏʀᴇ ᴛʜᴇ ʟᴀʀɢᴇsᴛ 4-ᴅɪɢɪᴛ ɴᴜᴍʙᴇʀ ʟᴇᴀᴠᴇs ᴀ ʀᴇᴍᴀɪɴᴅᴇʀ 3 = 9828 + 3

                                                                                                     = 9831.

(5) 

ғɪʀsᴛ, ᴡᴇ ɴᴇᴇᴅ ᴛᴏ ғɪɴᴅ ᴛʜᴇ ʟᴄᴍ ᴏғ 5,6,4 ᴀɴᴅ 3.

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ 5 = 5

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ 6 = 2 * 3

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ  4 = 2 * 2

ᴘʀɪᴍᴇ ғᴀᴄᴛᴏʀɪᴢᴀᴛɪᴏɴ ᴏғ 3 = 3

ʟᴄᴍ(5,6,4,3) = 2 * 2 * 3 * 5

                      = 60.

ɪɴ ᴏʀᴅᴇʀ ᴛᴏ ғɪɴᴅ ᴛʜᴇ ʀᴇᴍᴀɪɴᴅᴇʀ, ᴡᴇ sʜᴏᴜʟᴅ ᴅɪᴠɪᴅᴇ 2497 ᴡɪᴛʜ ᴛʜᴇ ʟᴄᴍ.

60) 2497( 41

      240

      ---------
 
            97

            60

      -----------

               37.

ᴛʜᴇʀᴇғᴏʀᴇ ᴛʜᴇ ʟᴇᴀsᴛ ɴᴜᴍʙᴇʀ ᴡʜɪᴄʜ sʜᴏᴜʟᴅ ʙᴇ ᴀᴅᴅᴇᴅ = (60 - 37)

                                                                                        = 23.

ʜᴏᴘᴇ ᴛʜɪs ʜᴇʟᴘs!