Other tags that relate to Gosh Cosh
Exponential function. Maximise/minimise/optimise. Indices. Graph plotters. Hyperbolic functions. Transformation of functions. Graphs. Mathematical induction. Mathematical reasoning & proof. Inequalities.There are 32 results
Broad Topics > Algebraic expressions, equations and formulae > Indices
Sums of Squares
Age 16 to 18 Challenge Level:
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?
Cube Roots
Age 16 to 18 Challenge Level:
Evaluate without a calculator: (5 sqrt2 + 7)^{1/3} - (5 sqrt2 - 7)^1/3}.
Modulus Arithmetic and a Solution to Dirisibly Yours
Age 16 to 18
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.
Negative Power
Age 14 to 16 Challenge Level:
What does this number mean ? Which order of 1, 2, 3 and 4 makes the highest value ? Which makes the lowest ?
Giants
Age 16 to 18 Challenge Level:
Which is the bigger, 9^10 or 10^9 ? Which is the bigger, 99^100 or 100^99 ?
Remainder Hunt
Age 16 to 18 Challenge Level:
What are the possible remainders when the 100-th power of an integer is divided by 125?
Power Up
Age 16 to 18 Challenge Level:
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
Big, Bigger, Biggest
Age 16 to 18 Challenge Level:
Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$?
Learn about Number Bases
Age 11 to 18
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.
Staircase
Age 16 to 18 Challenge Level:
Solving the equation x^3 = 3 is easy but what about solving equations with a 'staircase' of powers?
Sums of Squares and Sums of Cubes
Age 16 to 18
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 of two or more cubes.
Power Quady
Age 16 to 18 Challenge Level:
Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.
Novemberish
Age 14 to 16 Challenge Level:
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.
Climbing Powers
Age 16 to 18 Challenge Level:
$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or. . . .
The Public Key
Age 16 to 18 Challenge Level:
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.
Public Key Cryptography
Age 16 to 18
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.
Powerful Factors
Age 16 to 18 Challenge Level:
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.
Perfectly Square
Age 14 to 16 Challenge Level:
The sums of the squares of three related numbers is also a perfect square - can you explain why?
Pythagoras Mod 5
Age 16 to 18 Challenge Level:
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.
Enriching Experience
Age 14 to 16 Challenge Level:
Find the five distinct digits N, R, I, C and H in the following nomogram
Rachel's Problem
Age 14 to 16 Challenge Level:
Is it true that $99^n$ has 2n digits and $999^n$ has 3n digits? Investigate!
Tens
Age 16 to 18 Challenge Level:
When is $7^n + 3^n$ a multiple of 10? Can you prove the result by two different methods?
Card Shuffle
Age 11 to 16
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.
A Biggy
Age 14 to 16 Challenge Level:
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.
Growing
Age 16 to 18 Challenge Level:
Which is larger: (a) 1.000001^{1000000} or 2? (b) 100^{300} or 300! (i.e.factorial 300)
Telescoping Series
Age 16 to 18 Challenge Level:
Find $S_r = 1^r + 2^r + 3^r + ... + n^r$ where r is any fixed positive integer in terms of $S_1, S_2, ... S_{r-1}$.
Multiplication Magic
Age 14 to 16 Challenge Level:
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). . . .
Really Mr. Bond
Age 14 to 16 Challenge Level:
115^2 = (110 x 120) + 25, that is 13225 895^2 = (890 x 900) + 25, that is 801025 Can you explain what is happening and generalise?
NRICH is part of the family of activities in the Millennium Mathematics Project.