Provide an appropriate response.A student has noticed that every squared natural number is either a multiple of 4 or one larger than a multiple of 4, but isn't sure if this is always true. Use algebraic reasoning to explain to the student why this is always true.
What will be an ideal response?
Every even natural number is in the form of 2k and every odd natural number is in the form of 2k + 1, where k is some natural number.
Even natural numbers squared = (2k)2 = 4k2, which is in fact a factor of 4.
Odd natural numbers squared = (2k + 1)2 = 4k2 + 4k + 1 = 4(k2 + k) + 1, which is in fact one larger than a multiple of 4.
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