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 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?

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.

If a number is divisible by both 9 and 4, it must be divisible by 36.

If a number is divisible by both 6 and 4, it must be divisible by 24.

When you add an odd number to an even number we get a multiple of 6.

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

Using this shortcut, find out whether the following numbers are divisible by 11. Further, find the remainder if the number is not divisible by 11.

Write the digital roots of any 12 consecutive numbers. What do you observe?

Find the digital roots of some consecutive multiples of (i) 3, (ii) 4, and (iii) 6. What do you notice?

What are the digital roots of numbers that are 1 more than a multiple of 6? What do you notice?