Note: Unit digit in `(3127)^173` = Unit digit in `(7)^173` Now, `7^4` gives unit digit 1. `:. 7^172 = (7^6)^43` gives unit digit 1. `:. 7^173` gives unit digit=`(1 xx 7)=7`.

Note: ` 9 xx 6 xx 4=216`. In order to obtain 2 at the unit place it must be multiplied by a number whose unit digit is 7. So, * must be replaced by 7.

Note: Unit digit in `7^4` is 1. `:.` Unit digit in `7^68` is 1. `:.` Unit digit in `7^71` is 3. [`1 xx 7 xx 7 xx 7` gives unit digit 3] Again, every power of 6 will give unit digit 6. `:.` Unit digit in `6^59` is 6. Clearly, unit digit in `3^4` is 1. So, unit digit in `3^64` is 1. `:.` Unit digit in `3^65` is 3. `:.` Unit digit in `(7^71 xx 6^59 xx 3^65)` is 4. [`3 xx 6 xx 3` gives unit digit 4]

Note: Not available

Note: Required number `= 35999-(17102+8799) =35999-25901 =10089`

Note: `987 xx 999` `=987 xx (1000 - 1)` `= (987 xx 1000)-(987 xx 1)` `= (987000-987)` `=986013`

Note: `987 xx 999` `=987 xx (1000 - 1)` `= (987 xx 1000)-(987 xx 1)` `= (987000-987)` `=986013`

Note: `469157 xx 9999` `= 4691570000-469157` `=491100843`.

Note: `935421 xx 625` `= 935421 xx 25 xx 25` `= 935421 xx 100/4 xx 100/4` `= 9354210000/16` `=584638125`

Note: Use of formula, `( a^2 + b^2 + 2ab) = (a+b)^2` We have, `387 xx 387 + 114 xx 114 + 2 xx 387 xx 114` `=(387)^2 + (114)^2 + 2 xx 387 xx 114` `=(387 + 114)^2 =(501)^2 = (500 + 1)^2` `=(500)^2 + (1)^2 =2 xx 500 xx 1` `=250000 + 1 +1000 = 251001`