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Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

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N000ughty Thoughts

Factorial one hundred (written 100!) has 24 noughts when written in full and that 1000! has 249 noughts? Convince yourself that the above is true. Perhaps your methodology will help you find the number of noughts in 10 000! and 100 000! or even 1 000 000!

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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.


Stage: 4 and 5 Challenge Level: Challenge Level:1

Why do this problem?

As a simple exercise using modulus arithmetic where learners can make and test their own conjectures. By trying small values of n learners may make their own conjectures about when the expression is divisible by 6 and when it is not.

Key question

How can we write the expression as an equivalent expression modulo 3?