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More Mods

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

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Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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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.

Days and Dates

Stage: 4 Challenge Level: Challenge Level:1

Circle with numbers 0 to 6.

If today is Monday we know that in 702 days time (that is in 100 weeks and 2 days time) it will be Wednesday. This is an example of "clock" or "modular" arithmetic.

What day will it be in 15 days? 26 days? 234 days?

In 2, 9, 16 and 23 days from now, it will be a Wednesday. What other numbers of days from now will be Wednesdays? Can you generalise what you have noticed?

Choose a pair of numbers and find the remainders when you divide by 7. Find the remainder when you divide the total by 7. For example

$15 \div 7 = 2$ remainder $1$

$26 \div 7 = 3$ remainder $5$

$15 + 26 = 41$

$41 \div 7 = 5$ remainder $6$

Try some more numbers. What do you notice about the remainders? Can you justify what you see?

Now find the remainder when you divide the product of 15 and 26 by 7. What happens? Can you justify what you see? Try some more numbers.

How can I use these ideas to work out what day my birthday will be on next year?