Combinatorics | Number Theory Answers

Part 1: Prove that the system of linear congruences in one variable
x ≡ b1 mod m1
x ≡ b2 mod m2
is solvable if and only if (m1, m2)|b1 − b2. In this case prove that the solution is unique modulo [m1, m2].
Hint: follow the proof of the Chinese Remainder Theorem.
Part 2: Solve the system of congruences
x ≡ 3 mod 4
x ≡ 1 mod 6

To check whether a number is divisible by 19 one can repeatedly add twice the last digit
to the rest of the number. For example:
13547→ 1354 + 7 ∗ 2 = 1368 → 136 + 8 ∗ 2 = 152 → 15 + 4 = 19
thus 13547 is divisible by 19 (indeed, 13547=19*713). Give a proof of this divisibility rule.

Prove that there are infinitely many prime numbers of the form 6n + 5.
Hint: any prime number p > 3 has the form p = 6n + 1 or p = 6n + 5 for some integer n. Use this fact
and the fact that the product of two numbers of the form 6n + 1 has the same form. Also, read again
the proof of Proposition 1.22, where we proved that there exist infinitely many prime numbers of the
form 4n + 3.

Let 'a' and 'b' be relatively prime integers. Find all values of (a + 2b, a − 2b).

50 of the Students were asked their subject combination 18 offered mathematics, 21.
offered Chemistry 16 offered biology& chemistry
offered mathematics and Chemisttry 8 students offered maths and biology
9 offered Chemistry and biology, While 5
offered the three sublect combination to
the venn diagram find
The numbed of student that passes mathematics
The number of studeats that affred Chemistry
but offered neither father mothe nor blology
The number of students that offered biology
but offered neither further maths nor Chemistry
The number of students who did not offer
any of the three Subject Combination.
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In how many ways could you arrange a display of stationery supplies consisting of 14 notebooks, 5 reams of lined paper, and 50 pens if all the items are laid out in a row?

Let a and b be integers. Prove that (a, b) = 1 if and only if (a + b, ab) = 1.

Let a and b be relatively prime integers. Find all values of (a + 2b, a − 2b).

A researcher wishes to study railroad accidents he wishes to say like three rows from 10 class one railroads to rail road from six close to real world and one rail road for five class III real rose how many different possibilities are there for his study

Find the greatest number of four digits when divided by 3,5,7,9 leaves remainder 1,3,5,7 respectively