Math Detours · Grade 9 · Around the corner · 1

Proof with letters

Call an odd number 2n + 1 and you can prove things about every odd number at once. The sum of two odds, the square of an odd, and a calendar trick.

Name ____________________Date ____________

Tonight's idea

Add two odd numbers: \(3 + 5 = 8\), \(7 + 11 = 18\). Even each time. But checking examples never finishes: the odd numbers never run out. A letter finishes the job. An even number is two times a whole number: \(2n\). An odd number is one more: \(2n + 1\). So \(101 = 2(50) + 1\).

An odd number is a row of pairs with one left over. Add two of them and the leftovers make a pair.

Take any two odd numbers, \(2a + 1\) and \(2b + 1\). Two letters, because the numbers need not be equal. Their sum is

\[(2a + 1) + (2b + 1) = 2a + 2b + 2 = 2(a + b + 1).\]

Two times a whole number, so even. The letters stood for every pair of odd numbers at once. That is a proof: an argument that leaves no case out. An odd square: \((2n + 1)^2 = 4n^2 + 4n + 1 = 4(n^2 + n) + 1\). Always one more than a multiple of 4.

To prove a fact about every number: write the number with a letter, do the algebra, then read what the result says.

Watch one: a calendar trick

Box a 2 by 2 square of dates, say 10, 11 above 17, 18. Multiply the diagonals: \(10 \times 18 = 180\) and \(11 \times 17 = 187\). They differ by 7. Luck?

Call the top-left date \(d\). The others are \(d + 1\), \(d + 7\) and \(d + 8\), since one row down adds 7.

\(d(d + 8) = d^2 + 8d\) and \((d + 1)(d + 7) = d^2 + 8d + 7\). The difference is 7 for every box, every month.

Where you met this: the 3 by 3 box in Calendar magic.

Your turn

Use \(2a\) for an even number and \(2b + 1\) for an odd one. Prove that (a) an even number plus an odd number is odd; (b) an odd number times an odd number is odd.
Three whole numbers in a row can be written \(n\), \(n + 1\), \(n + 2\). (a) Prove that their sum is always a multiple of 3. (b) Is the sum of four whole numbers in a row always a multiple of 4? Use letters to decide.
Box a 3 by 3 square of dates and call the middle date \(m\). (a) Write the four corner dates using \(m\). (b) Multiply each pair of opposite corners. Prove that the two products always differ by the same amount, and find it. (c) Check with the box whose middle date is 17.
A two-digit number with tens digit \(a\) and units digit \(b\) is worth \(10a + b\). Reverse the digits to get a second number. Prove that (a) the difference of the two numbers is always a multiple of 9; (b) their sum is always a multiple of 11. (c) Check both with 72.

Stretch: one more than a multiple of 8

Square some odd numbers and subtract 1: you get 8, 24, 48, 80. All multiples of 8, not just of 4. Prove that this always happens. Start from \((2n + 1)^2 = 4n(n + 1) + 1\) and ask what you know about \(n(n + 1)\).

Answers

(a) \(2a + (2b + 1) = 2(a + b) + 1\): one more than an even number, so odd. (b) \((2a + 1)(2b + 1) = 4ab + 2a + 2b + 1 = 2(2ab + a + b) + 1\): odd. (Here both odd numbers need their own letter.)
(a) \(n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)\), three times a whole number. (b) No. \(n + (n + 1) + (n + 2) + (n + 3) = 4n + 6 = 4(n + 1) + 2\), which is always 2 more than a multiple of 4. So it is never a multiple of 4.
(a) \(m - 8\), \(m - 6\), \(m + 6\), \(m + 8\). (b) \((m - 8)(m + 8) = m^2 - 64\) and \((m - 6)(m + 6) = m^2 - 36\). The difference is \(64 - 36 = 28\) every time. (c) Corners 9, 11, 23, 25: \(9 \times 25 = 225\) and \(11 \times 23 = 253\), and \(253 - 225 = 28\).
The reversed number is \(10b + a\). (a) \((10a + b) - (10b + a) = 9a - 9b = 9(a - b)\). (b) \((10a + b) + (10b + a) = 11a + 11b = 11(a + b)\). (c) \(72 - 27 = 45 = 9 \times 5\) and \(72 + 27 = 99 = 11 \times 9\).

Stretch. \(n\) and \(n + 1\) are next to each other, so one of them is even, and \(n(n + 1)\) is even. Write \(n(n + 1) = 2k\). Then \((2n + 1)^2 = 4(2k) + 1 = 8k + 1\), one more than a multiple of 8.

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