16. Prove that an integer (a,-1an-2 · . . ao)10 is divisible by 11 if and only if ao + a2 + a4 + . . . = aj + a3 + a5 + · ·. (mod 11). [Hint: 10 = -1 (mod 11).] %3D

Q: 23. For any real number a, a ‡ 0, show that a +1 a +4 a +7 a +5 a+6 a +9 a -2 a +3 and a +8 | = 0, a…

A:

Q: Find a positive integer less than 8 for x that satisfies x = 720 + 3 (mod 7).

A:

Q: Use the definitions  to show that the following statements are true. For example, to show that 5 |…

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Q: It is given that (²1) = 4, (3) k 2 -(" + ²) Write and expression for = B, (x+1)= = C (where n, k, A,…

A: Use the properties of combination.

Q: 4x=22(mod10)

A:

Q: Prove that 5 does not divide 5n+3 for any integer n.

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Q: Prove that a number in the form 6k+5,” what I mean is a number in the form of 6 times some integer…

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Q: (b) Prove that any odd number is expressible in the form 4n + 1 or 4n + 3 where n is an integer.

A: Number System

Q: 11.) What is an equivalent form of 4 5+i -?

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Q: Prove that an odd number multiplied by an odd number is odd.

A: The objective is to prove that an odd number multiplied by an odd number is odd.

Q: 2 4. Express 989 = 23 43 in the form x² + xy + 5y².

A: The given number is 989. The objective is to express the given number in the form of x2+xy+5y2.

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: 6. Letn be a positive integer. Prove that 1+ 4+7++ (3n − 2) = = 3n² 2

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Q: Deduce from Q2 that if an integer N can be written as N = x² + ny² , and if a prime number q | N can…

A: We will use standard division algorithms to deduce the given statement.

Q: x² - 2x = 8(mod19) X= (mod19)

A:

Q: Show that if k is a positive integer, then (15k+4)/(10k+3) is in lowest terms.

A: We use the concept of Ring theory that a fraction is in lowest form if the gcd of numerator and…

Q: Solve s(4- dx 3x2

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Q: √38 10 V √ 65

A: To check whether the given triangle is right triangle or not.

Q: Let n be an integer, use the definition of even to prove that 10n+ 4 is even. 10n+4=0x0 Since is a…

A: Let n be an integer. To prove that for any integer n the number 10n+4 is even. An even integer is…

Q: 5x² + 3x = 3(mod17) X = (mod17)

A: Given that 5x2+3x≡3 (mod 17)x(5x+3)≡3 (mod 17)x≡3 (mod 17)  or   (5x+3)≡3 (mod 17)x≡3 (mod 17)  or…