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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.

Transposition Fix

Stage: 4 Challenge Level: Challenge Level:1

An ISBN Number

Suppose an operator types a US Bank check code into a machine and transposes two adjacent digits (i.e. swaps the order of two adjacent digits) for example the operator types $a_3a_2$ instead of $a_2a_3$, will the machine pick up every error of this type? Does the same apply to ISBN numbers; will a machine detect transposition errors in these numbers?

US Bank check codes are nine digit identification numbers $a_1a_2\cdots a_{9}$ using the digits 0 to 9 where
$7a_1 + 3a_2 + 9a_3 + 7a_4 + 3a_5 + 9a_6 + 7a_7 + 3a_8 \equiv a_9$ mod 10.

ISBN Numbers have ten digits $a_1\cdots a_{10}$ using the digits 0 to 9 where
$10a_1+9a_2+8a_3+\cdots +3a_8+2a_9+a_{10} \equiv 0$ mod 11.