Question:Find the unit's digit in the product `(256 xx 27 xx 159 xx 182)`. 

Product of unit's digits in given numbers `= ( 6 xx 7 xx 9 xx 2) = 756`
`:.` Unit digit in the given product is `6`.

Question:Find the unit's digit in the product `( 3^67 xx 6^39 xx 7^53 )` 

Clearly, unit digit in `3^4` is 1.
∴ Unit digit in `3^64` is 1.
∴ Unit digit in `3^67` is 7. [Because Unit digit in `1 xx 3 xx 3 xx 3` is `7` ]
Clearly, unit digit in every power of 6 is 6
∴ Unit digit in `6^39` is 6.
Clearly, unit digit in `7^4` is 1.
∴ Unit digit in `7^52` is 1.
∴ Unit digit in `7^53` is 7. [Because Unit digit in `1 xx 7` is 7 ]
∴ Unit digit in given product = Unit digit in `( 7 xx 6 xx 7 ) = 4`.

Question:Prime numbers less than `100` 

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Test : Let p be a given number. Find a whole number nearly greater than the square root of p.
Let `k > sqrt p` .
Test whether p is divisible by any prime number less than k. If yes, then p is not prime. Otherwise, p is prime.

Question:Which of the following numbers is not divisible by `2`?
`3567516, 9130525, 7870823` and `1371594` 

Clearly, the unit's digit of `9130525` is `5`, which is not divisible by `2`. 
So, `9130525` is not divisible by `2`.
Each one of the other numbers has a unit divisible by `2`.
So, each one other than `9130525` is divisible by `2`.

Question:Which of the following numbers is divisible by 3 ?
 1) 541326, 2) 5967013 

1) Sum of digits in 541326 is 21, which is divisible by 3.

 So, 541326 is divisible by 3

 2) Sum of digits in 5967013 is 31, which is not divisible by 3.
 
 Hence, 5967013 is not divisible by 3.

Question:Which of the following numbers is divisible by 9 ?
i) 19725462 ii) 36870521 

i) Sum of digits in 19725462 is 36, which is divisible by 9.
So, the given number is divisible by 9.
ii) Sum of digits in 36870521 is 32, which is not divisible by 9.
So, the given number is not divisible by 9.

Question:Which of the following numbers is divisible by 4 ?
i) 67920594 ii) 618703572 

i) The number formed by the last 2 digits in the given number is 94, which is not divisible by 4
`:.` 67920594 is not divisible by 4.
ii) The number formed by the last 2 digits in the given number is 72, which is not divisible by 4
`:.` 618703572 is divisible by 4.

Question:Which of the following numbers is divisible by 8 ?
i) 98016542 ii) 106598304 

i) The number formed by the last 3 digits of the given number is 542, which is not divisible by 8.
`:.` 98016542 is not divisible by 8.
ii) The number formed by the last 3 digits of the given number is 304, which is not divisible by 8.
`:.` 106598304 is divisible by 8.

Question:Show that 4832718 is divisible by 11. 

(Sum of digits at odd places) - (Sum of digits at even places)
=[(8 + 7 + 3 + 4) - (1 + 2 + 8)]=11, which is divisible by 11.
`:.` 4832718 is divisible by 11.

Question:If a number is divisible by both 4 and 6, is it always divisible by 24 ? Why ? Give an example 

No, since 6 and 4 are not co-prime.
36 is divisible by 6 as well as 4 but it is not divisible by 24.