Divisibility and remainders
Types of exercises: remainders when dividing integers, divisibility for every integer, a gap in a proof about divisibility and remainders.
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Problem
What remainders can the square of an integer leave when divided by 4?
Which of these numbers divides it for every integer n?
(3n + 5)² − (3n − 5)²
What number should go in the box ☐ for the reasoning to be correct?
An integer n leaves a remainder of 4 when divided by 5, so n = 5k + 4 for some integer k. Hence: n² = (5k + 4)² = ☐k² + 40k + 16 = 5(5k² + 8k + 3) + 1 The first term is divisible by 5, and the last one is a natural number less than 5, so it is the remainder when n² is divided by 5.
Problem
A natural number n leaves a remainder of 6 when divided by 9. What remainder does n(n + 1) leave when divided by 9?
Which of these numbers divides it for every integer n?
(n + 2)² − (n − 11)²
What number should go in the box ☐ for the reasoning to be correct?
An integer n leaves a remainder of 2 when divided by 5, so n = 5k + 2 for some integer k. Hence: n² = (5k + 2)² = 25k² + ☐k + 4 = 5(5k² + 4k) + 4 The first term is divisible by 5, and the last one is a natural number less than 5, so it is the remainder when n² is divided by 5.
Problem
A natural number n leaves a remainder of 2 when divided by 5. What remainder does n² + 5 leave when divided by 5?
Which of these numbers divides it for every integer n?
6n² + 6n
What number should go in the box ☐ for the reasoning to be correct?
An integer n leaves a remainder of 1 when divided by 3, so n = 3k + 1 for some integer k. Hence: n² = (3k + 1)² = 9k² + 6k + 1 = 3(3k² + ☐k) + 1 The first term is divisible by 3, and the last one is a natural number less than 3, so it is the remainder when n² is divided by 3.
Problem
A natural number n leaves a remainder of 1 when divided by 8. What remainder does 6n² leave when divided by 8?
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