Number Play

The sum of four consecutive numbers is 34. What are these numbers?

Suppose p is the greatest of five consecutive numbers. Describe the other four numbers in terms of p.

Find a few numbers that leave a remainder of 2 when divided by 3 and a remainder of 2 when divided by 4. Write an algebraic expression to describe all such numbers.

I hold some pebbles, not too many, When I group them in 3's, one stays with me. Try pairing them up — it simply won't do, A stubborn odd pebble remains in my view. Group them by 5, yet one's still around, But grouping by seven, perfection is found. More than one hundred would be far too bold, Can you tell me the number of pebbles I hold?

Tathagat has written several numbers that leave a remainder of 2 when divided by 6. He claims, 'If you add any three such numbers, the sum will always be a multiple of 6.' Is Tathagat's claim true?

When divided by 7, the number 661 leaves a remainder of 3, and 4779 leaves a remainder of 5. Without calculating, can you say what remainders the following expressions will leave when divided by 7? Show the solution both algebraically and visually.

Find a number that leaves a remainder of 2 when divided by 3, a remainder of 3 when divided by 4, and a remainder of 4 when divided by 5. What is the smallest such number? Can you give a simple explanation of why it is the smallest?

Find, without dividing, whether the following numbers are divisible by 9.

Find the smallest multiple of 9 with no odd digits.

Find the multiple of 9 that is closest to the number 6000.

How many multiples of 9 are there between the numbers 4300 and 4400?

Take any 4 consecutive numbers. For example, 3, 4, 5, and 6. Place '+' and '–' signs in between the numbers. How many different possibilities exist? Write all of them.

Without computing them, find out which of the following arithmetic expressions are even: 43 + 37, 672 - 348, $4 \times 347$ × 3, 708 - 477, 809 + 214, $119 \times 303$, 543 - 479, 513³

Using our understanding of how parity behaves under different operations, identify which of the following algebraic expressions give an even number for any integer values for the letter-numbers: 2a + 2b, 3g + 5h, 4m + 2n, 2u - 4v, 13k - 5k, 6m - 3n, x² + 2, b² + 1, 4k × 3j

Take a pair of even numbers. Add them. Is the sum divisible by 4?

Examine each of the following statements, and determine whether it is 'Always true', 'Sometimes true', 'Never true'. Statement 6: If a number is divisible by both 9 and 4, it must be divisible by 36.

Examine each of the following statements, and determine whether it is 'Always true', 'Sometimes true', 'Never true'. Statement 7: If a number is divisible by both 6 and 4, it must be divisible by 24.

Find a number that has a remainder of 3 when divided by 5. Write more such numbers.

Which algebraic expression(s) capture all numbers that leave a remainder of 3 when divided by 5? (i) 3k + 5 (ii) 3k - 5 (iii) 3k/5 (iv) 5k + 3 (v) 5k - 2 (vi) 5k - 3

Can you say, without actually calculating, which of these numbers are divisible by 9: 999, 909, 900, 90, 990?