We have: 

⇒a=2017n−2016b⇒a=2017n−2016b

We have: 2a+2015b=2(2017n−2016b)+2015b=2.2017n−2017b⋮2017

We have: 

⇒a=2017n−2016b⇒a=2017n−2016b

We have: 2a+2015b=2(2017n−2016b)+2015b=2.2017n−2017b⋮20172a+2015b=2(2017n−2016b)+2015b=2.2017n−2017b⋮2017

Same as above. We hava: ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩3a+2014b=3.2017n−2.2017b⋮20174a+2013b=4.2017n−3.2017b⋮2017...2015a+2b=2015.2017n−2014.2017b{3a+2014b=3.2017n−2.2017b⋮20174a+2013b=4.2017n−3.2017b⋮2017...2015a+2b=2015.2017n−2014.2017b

So that: A⋮2017.2017...2017A⋮2017.2017...2017

⇒A⋮20172014

We have : 100 + 500 : 5 - 40 

= 100 + 100 - 40

= 200 - 40

= 160 

If p = 2 then p + 8 isn't a prime number.

If p = 3 then p + 8 and p + 16 are prime numbers.

  ⇒32+(3+8)2+(3+16)2=491⇒32+(3+8)2+(3+16)2=491

If p > 3 then [p=3k+1p=3k+2[p=3k+1p=3k+2

  + If p = 3k + 1 then p + 8 = 3k + 9=3(k+1) isn't a prime number.

  + If p = 3k + 2 then p + 16 = 3k + 18=3(k+6) isn't a prime number.

Ans: 491.

a + b + c = 72, then there's an even number and two odd numbers.

The only even prime number is 2, so a = 2.

$b+c=70=3+67=7+61=11+59=17+53=23+47=29+41$Ans : a = 2 and $\left(b;c\right)\in \{\left(3;67\right);\left(67;3\right);\left(7;61\right);\left(61;7\right);\left(11;59\right);\left(59;11\right);\left(17;53\right);\left(53;17\right);\left(23;47\right);\left(47;23\right);\left(29;41\right);\left(41;29\right)\}$

From 7m + 5n = 31, we can see that 7m or 5n is even.

- If 7m is even, m is even and prime => m = 2.

=> 5n = 31 - 7 x 2 = 17. n ⋮̸⋮̸ 5, so no numbers sastify the problem.

- If 5n is even, n is even and prime => n = 2.

=> 7m = 31 - 5 x 2 = 21 => m = 3 is a prime number.

The value of: mn + nm = 32 + 23 = 9 + 8 = 17.

We have: 

$\Rightarrow a=2017n-2016b$

We have: $2a+2015b=2\left(2017n-2016b\right)+2015b=2.2017n-2017b⋮2017$

Same as above. We hava: $\{\begin{array}{c}3a+2014b=3.2017n-2.2017b⋮2017\\ 4a+2013b=4.2017n-3.2017b⋮2017\\ ...\\ 2015a+2b=2015.2017n-2014.2017b\end{array}$

So that: $A⋮\mathrm{2017.2017...2017}$

$\Rightarrow A⋮{2017}^{2014}$

We have : A = 1.2 + 2.3 + 3.4 + ...... + 100.101

    =>       3A = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ....... + 99.100.101 

    =>       3A = 99.100.101

    =>         A = 99.100.101/3

    =>         A = 333300

B = 1 + 2 + 3 + ...... + 100

B is numbered as:

           100 - 1 + 1 = 100 (số)

Total B is:

           (100 x 101) : 2 = 5050 

Put A = 1.3 + 3.5 + 5.7 + ...... + 99.101

=> 4A = 1.3.(5 - 1) + 3.5.(7 - 3) + ...... + 99.101.(103 - 99)

=> 4A = 1.3.5 - 1.3.5 + 3.5.7 - 3.5.7 + ...... + 99.100.101

=> 4A = 99.100.101

=> A = 99.100.101/4

=> A = 249975

Put A = 1.3 + 3.5 + 5.7 + ...... + 99.101

=> 4A = 1.3.(5 - 1) + 3.5.(7 - 3) + ...... + 99.101.(103 - 99)

=> 4A = 1.3.5 - 1.3.5 + 3.5.7 - 3.5.7 + ...... + 99.100.101

=> 4A = 99.100.101

=> A = 99.100.101/4

=> A = 249975

Put : A = 1.3 + 3.5 + 5.7 + ...... + 99.101

=> 4A = 1.3.(5 - 1) + 3.5.(7 - 3) + ...... + 99.101.(103 - 99)

=> 4A = 1.3.5 - 1.3.5 + 3.5.7 - 3.5.7 + ...... + 99.100.101

=> 4A = 99.100.101

=> A = 99.100.101/4

=> A = 249975

Put A = 1.2 + 2.3 + 3.4 + ....... + 199.200

 => 3A = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ....... + 199.200.201

=> 3A = 199.200.201

=> A = 199.200.201/3

=> A = 2666600

Put A = 1.2 + 2.3 + 3.4 + ....... + 199.200

 => 3A = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ....... + 199.200.201

=> 3A = 199.200.201

=> A = 199.200.201/3

=> A = 2666600

Put A = 1.2 + 2.3 + 3.4 + ....... + 199.200

 => 3A = 1.2.3 - 1.2.3 + 2.3.4 - 2.3.4 + ....... + 199.200.201

=> 3A = 199.200.201

=> A = 199.200.201/3

=> A = 2666600

That worker makes 3 tools out of time is:

                          12 hours - 7 hours 30 minute = 4 hours 30 minute = 4,5 hours

The worker makes 1 product time out is :  

                          4,5 : 3 = 1,5 (hours)

                                 Answer : 1,5 hours

That worker makes 3 tools out of time is:

                          12 hours - 7 hours 30 minute = 4 hours 30 minute = 4,5 hours

The worker makes 1 product time out is :  

                          4,5 : 3 = 1,5 (hours)

                                 Answer : 1,5 hours