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Adding All Nine

Make a set of numbers that use all the digits from 1 to 9, once and once only. Add them up. The result is divisible by 9. Add each of the digits in the new number. What is their sum? Now try some other possibilities for yourself!

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Double Digit

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. Try lots of examples. What happens? Can you explain it?

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Legs Eleven

Take any four digit number. Move the first digit to the 'back of the queue' and move the rest along. Now add your two numbers. What properties do your answers always have?

Quiz Questions

Stage: 3 Short Challenge Level: Challenge Level:1

Jack must get at least $13$ questions correct in order to score enough points. $13 \times 7 = 91$, so these questions score him $91$ points. Then he needs two wrong answers to reduce his score to $87$, meaning $5$ questions are left unanswered.

If he gets more than $13$ questions correct, marks can only be deducted in twos. Therefore he must get an odd number of marks from the correct questions, so must score at least $15 \times 7 = 105$ points from them. This means he needs at least $9$ incorrect questions also, a total of at least $15+9=24$. However, there are only $20$ questions in the test, so this cannot happen.

Therefore, the only way to get $87$ is to get $13$ questions correct and $2$ wrong, leaving $5$ questions unattempted.

This problem is taken from the UKMT Mathematical Challenges.
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