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Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

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Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

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A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

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Can you create a Latin Square from multiples of a six digit number?

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Visitors to Earth from the distant planet of Zub-Zorna were amazed when they found out that when the digits in this multiplication were reversed, the answer was the same! Find a way to explain. . . .

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Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

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Choose two digits and arrange them to make two double-digit numbers. Now add your double-digit numbers. Now add your single digit numbers. Divide your double-digit answer by your single-digit answer. . . .

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Each letter represents a different positive digit AHHAAH / JOKE = HA What are the values of each of the letters?

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Find the values of the nine letters in the sum: FOOT + BALL = GAME

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Can you show that 1^99 + 2^99 + 3^99 + 4^99 + 5^99 is divisible by 5?

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Replace the letters with numbers to make the addition work out correctly. R E A D + T H I S = P A G E

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Can you arrange the digits 1,2,3,4,5,6,7,8,9 into three 3-digit numbers such that their total is close to 1500?

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What is the sum of all the digits in all the integers from one to one million?

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In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

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The letters in the following addition sum represent the digits 1 ... 9. If A=3 and D=2, what number is represented by "CAYLEY"?

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Explore the factors of the numbers which are written as 10101 in different number bases. Prove that the numbers 10201, 11011 and 10101 are composite in any base.

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By selecting digits for an addition grid, what targets can you make?

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Explore the relationship between simple linear functions and their graphs.

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What happens when you add a three digit number to its reverse?

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Try out some calculations. Are you surprised by the results?

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This addition sum uses all ten digits 0, 1, 2...9 exactly once. Find the sum and show that the one you give is the only possibility.

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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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How many positive integers less than or equal to 4000 can be written down without using the digits 7, 8 or 9?

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When the number x 1 x x x is multiplied by 417 this gives the answer 9 x x x 0 5 7. Find the missing digits, each of which is represented by an "x" .

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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?

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Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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Watch our videos of multiplication methods that you may not have met before. Can you make sense of them?

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How many solutions can you find to this sum? Each of the different letters stands for a different number.

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A composite number is one that is neither prime nor 1. Show that 10201 is composite in any base.

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A 2-Digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

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Can you explain the surprising results Jo found when she calculated the difference between square numbers?

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Think of a two digit number, reverse the digits, and add the numbers together. Something special happens...

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The Number Jumbler can always work out your chosen symbol. Can you work out how?

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How many more miles must the car travel before the numbers on the milometer and the trip meter contain the same digits in the same order?

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32 x 38 = 30 x 40 + 2 x 8; 34 x 36 = 30 x 40 + 4 x 6; 56 x 54 = 50 x 60 + 6 x 4; 73 x 77 = 70 x 80 + 3 x 7 Verify and generalise if possible.

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

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The number 3723(in base 10) is written as 123 in another base. What is that base?

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Pick the number of times a week that you eat chocolate. This number must be more than one but less than ten. Multiply this number by 2. Add 5 (for Sunday). Multiply by 50... Can you explain why it. . . .

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Think of two whole numbers under 10, and follow the steps. I can work out both your numbers very quickly. How?

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Where should you start, if you want to finish back where you started?

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Suppose you had to begin the never ending task of writing out the natural numbers: 1, 2, 3, 4, 5.... and so on. What would be the 1000th digit you would write down.

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Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your opponent.

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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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Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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How many six digit numbers are there which DO NOT contain a 5?

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Can you produce convincing arguments that a selection of statements about numbers are true?

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There are nasty versions of this dice game but we'll start with the nice ones...