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Learn about Number Bases

Stage: 5

We are used to writing numbers in base ten, using 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. Eg. 75 means 7 tens and five units. This article explains how numbers can be written in any number base.

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Rachel's Problem

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Is it true that 99^n has 2n digits and 999^n has 3n digits? Investigate!

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Card Shuffle

Stage: 3 and 4

This article for students and teachers tries to think about how long would it take someone to create every possible shuffle of a pack of cards, with surprising results.

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Remainder Hunt

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

What are the possible remainders when the 100-th power of an integer is divided by 125?

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Staircase

Stage: 5 Challenge Level: Challenge Level:1

Solving the equation x^3 = 3 is easy but what about solving equations with x to the power x^3 or x to the power x to the power x^3 or having a 'staircase' of powers?

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The Public Key

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Find 180 to the power 59 (mod 391) to crack the code. To find the secret number with a calculator we work with small numbers like 59 and 391 but very big numbers are used in the real world for this.

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More Mods

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

What is the units digit for the number 123^(456) ?

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Really Mr. Bond

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

115^2 = (11 x 12)x 25, that is 13225 895^2 = (89 x 90)x 25, that is 801025 Can you explain what is happening and generalise?

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Climbing Powers

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

2^3^4 could be (2^3)^4 or 2^(3^4). Does it make any difference? For both definitions, which is bigger r^r^r^r... where the powers of r go on for ever, or (r^r)^r, where r is the square root of 2?

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Tens

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

When is 7^n + 3^n a multiple of 10? Can you prove the result by two different methods?

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Sums of Squares

Stage: 5 Challenge Level: Challenge Level:1

Prove that 3 times the sum of 3 squares is the sum of 4 squares. Rather easier, can you prove that twice the sum of two squares always gives the sum of two squares?

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Negative Power

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?

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Enriching Experience

Stage: 4 Challenge Level: Challenge Level:1

Find the five distinct digits N, R, I, C and H in the following nomogram

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Multiplication Magic

Stage: 4 Challenge Level: Challenge Level:1

Given any 3 digit number you can use the given digits and name another number which is divisible by 37 (e.g. given 628 you say 628371 is divisible by 37 because you know that 6+3 = 2+7 = 8+1 = 9). . . .

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Elevens

Stage: 5 Challenge Level: Challenge Level:1

Add powers of 3 and powers of 7 and get multiples of 11.

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Telescoping Series

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Find S_r = 1^r + 2^r + 3^r + ... + n^r where r is any fixed positive integer in terms of S_1, S_2, ... S_{r-1}.

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Growing

Stage: 5 Challenge Level: Challenge Level:1

Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)

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Novemberish

Stage: 4 Challenge Level: Challenge Level:1

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

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Public Key Cryptography

Stage: 5

An introduction to the ideas of public key cryptography using small numbers to explain the process. In practice the numbers used are too large to factorise in a reasonable time.

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Giants

Stage: 4 and 5 Challenge Level: Challenge Level:2 Challenge Level:2

Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?

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Powerful Factors

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Use the fact that: x²-y² = (x-y)(x+y) and x³+y³ = (x+y) (x²-xy+y²) to find the highest power of 2 and the highest power of 3 which divide 5^{36}-1.

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More Sums of Squares

Stage: 5

Tom writes about expressing numbers as the sums of three squares.

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Pythagoras Mod 5

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.

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Perfectly Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

The sums of the squares of three related numbers is also a perfect square - can you explain why?

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Sums of Squares and Sums of Cubes

Stage: 5

An account of methods for finding whether or not a number can be written as the sum of two or more squares or as the sum orf two or more cubes.

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How Many?

Stage: 5 Challenge Level: Challenge Level:1

Find all the solutions to the this equation.

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Modulus Arithmetic and a Solution to Dirisibly Yours

Stage: 5

Peter Zimmerman from Mill Hill County High School in Barnet, London gives a neat proof that: 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.

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Power Quady

Stage: 4 Challenge Level: Challenge Level:1

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

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A Biggy

Stage: 4 Challenge Level: Challenge Level:1

Find the smallest positive integer N such that N/2 is a perfect cube, N/3 is a perfect fifth power and N/5 is a perfect seventh power.

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Cube Roots

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.

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Power Up

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x

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Big, Bigger, Biggest

Stage: 5 Challenge Level: Challenge Level:1

Which is the biggest and which the smallest of these numbers and by how do they compare in magnitude? A = 2000^{2002} B = 2001^{2001} C = 2002^{2000}