III) Are there positive integers x and y suct that x³ = 2² +15? (Hint: for every integer x, x³ 0, 1, 6 (mod 7) )

Q: How do I prove that if a, b, and c are integers with c not equal to 0 then ac = bc => a = b?

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Q: Prove the following statement: 22 +23+24+... 2n+1 = 2n+2-4

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Q: 7. Prove that 4 3m² – n² for all odd integers m and n.

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Q: Let "xy" denote "x does not divide y." Prove by any method. Let a and b be integers. If 3 ab, then…

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Q: Find all solutions of the congruence 8x ≡ 4 (mod 12)

A: Step 1: Divide by the greatest common divisor (GCD)The GCD of 8 and 12 is 4. We can divide both…

Q: Show that if x+yi=(a+bi)- is true, then x-yi=(a-bi)-is all real numbers). true (x, y, a, and b are

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Q: Consider all positive integers that are mueiples of 20 and that are less than or equal to 300. What…

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Q: Prove there are no natural numbers a and b such that √a+b = √a + √b.c

A: Assume such a,b exists and prove by contradiction

Q: 4. Prove that if m and n are positive integers and x is a real number, then %3D m

A: We know ⨽x⨼ = x if x is an integer. Here, since m and n are integers, we have the following steps:…

Q: 6: Prove by contradiction that for any positive integer x, x+ 49 x ≥ 14.

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Q: 10. Let x be a real number. Prove that at least one of the numbers /2 – x or /2 +x is irrational.

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Q: How many integer solutions to x1 + x2 + x3 + x4 x₁ ≥ 4, x2 ≥ 2, x3 ≥ 5 and x4 ≥ 3? > = 31 are there…

A: Find change all inequality to grater then equal to 0. Then use the theorem

Q: Use a proof by contrapositive to prove that, if (x + 12)/x is an integer, then x < 13.

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Q: x² + 4x = 10(mod11) (mod11) X=

A: Given that x2+4x≡10 (mod 11)x2+4x+4≡10+4 (mod 11)x2+4x+4≡14 (mod 11)x2+4x)(x+2)2≡14+11 (mod…

Q: True or False : 4x ≡ 12 (mod 10) ⟹ x ≡ 3 (mod 10)

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Q: 10. Let a and b be natural numbers that are co-prime. Prove that (b-a) and b must also be co-prime.

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Q: Prove that for all real numbers Ex)y= -(1u)

A: We have to prove that for all real numbers (- x)y= - (xy)

Q: x² + x = 13(mod17) X= * (mod17)

A:

Q: 2. For all real numbers a, b and c, prove that if a <b <c then a+b+c a < 3 < C.

A: given that a, b, c are real numbers we have to prove that if      a&lt;b&lt;c    then…

Q: Find all integer values of x for which 3x = 4 (mod 8).

A: given congruence equation 3x≡4(mod 8)

Q: Solve x2+3x+2 is congruent to 0 (mod 55)

A: 55=5×11 and gcd(5,11)=1, hence they are co-prime.The given system of equation x2+3x+2≡0mod 55 is…

Q: Prove that a1 + a2 = b₁ + b₂ mod n.

A: We have to prove that a1+a2=b1+b2 (mod n).Given:a1≡b1 (mod n)a2≡b2 (mod n)This means there exist…

Q: Suppose that b and c are integers where 9b = 12c. Then 9|c and 12|b.

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Q: only d)

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Q: Consider the following theorem: There do not exist three consecutive odd integers a, b and c such…

A: In the last , I have written the theorem " There does not exist three consecutive odd integers…