Algebra, Algebraic Techniques, Short Multiplication Formulas / Algebraic Identities
These are specific algebraic identities like `(a+b)^2 = a^2+2ab+b^2`, `(a-b)^2 = a^2-2ab+b^2`, and `a^2-b^2 = (a-b)(a+b)`. Questions involve recognizing when to apply these formulas for quick expansion, factorization, or simplification of expressions.
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Calculate without a calculator using the most efficient method:
\( { (2001 \cdot 2021+100) \cdot (1991 \cdot 2031+400) \over 2011^4}\)
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Answer without using pen and paper (and without a calculator, of course
):Which number is larger: `12345678^2` or `12345677*12345679` ?
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Prove that the difference of squares of two consecutive odd numbers is divisible by `8`.
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Prove that for every prime number `p>3 ` the following holds: `p^2-1` is divisible by `6`.
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Given a natural number `A`. When it is increased by `1`, its square increases by `1001`. Find `A`.
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Is the following number prime?
`4^9 + 6^10 + 3^20`
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The distinct real numbers `x` and `y` satisfy the equation:
`x^2 – 2000x = y^2 – 2000y`
Find the sum of `x` and `y`.
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Calculate the product:
`(1-1/4)(1-1/9)(1-1/16)*...*(1-1/225)`
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It is known that `a+b-c=6`, `ac+bc-ab=2`. Find `a^2+b^2+c^2`.
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Calculate the value of the expression (without a calculator):
`(2^3-1)/(2^3+1)*(3^3-1)/(3^3+1)*(4^3-1)/(4^3+1)*...*(100^3-1)/(100^3+1)`