A University-Level Understanding of Number with an eye towards elementary education

1.

1. \(\displaystyle 555 + 777\)
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2. \(\displaystyle 13 + 23\)
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3. \(\displaystyle 467 + 809\)
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4. \(\displaystyle 88701 + 4097\)
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2.

3.

4.

5.

1. \(\displaystyle 238_{ten} + 167_{ten}\)
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2. \(\displaystyle 132_{four} + 23_{four}\)
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3. \(\displaystyle 2B7_{twelve} + 59_{twelve}\)
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4. \(\displaystyle 675_{eight} + 47_{eight}\)
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6.

1. \(\displaystyle 152_{ten} + 146_{ten}\)
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2. \(\displaystyle 634_{ten} + 237_{ten}\)
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3. \(\displaystyle 245_{eight} + 367_{eight}\)
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4. \(\displaystyle 245_{twelve} + 367_{twelve}\)
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7.

1. \(\displaystyle 141_{ten} + 431_{ten}\)
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2. \(\displaystyle 416_{ten} + 235_{ten}\)
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3. \(\displaystyle 331_{four} + 231_{four}\)
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4. \(\displaystyle 472_{twelve} + 801_{twelve}\)
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5. \(\displaystyle 341_{eight} + 777_{eight}\)
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8.

1. \(\displaystyle 934_{ten} + 374_{ten}\)
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2. \(\displaystyle 799_{ten} + 110_{ten}\)
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3. \(\displaystyle 333_{four} + 320_{four}\)
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4. \(\displaystyle 8A9_{twelve} + BBB_{twelve}\)
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5. \(\displaystyle 751_{eight} + 27_{eight}\)
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9.

1. \(\displaystyle 1222_{ten} + 237_{ten}\)
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2. \(\displaystyle BBB_{twelve} + 437_{twelve}\)
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3. \(\displaystyle 307_{eight} + 561_{eight}\)
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10.

11.

1. When adding four addends, what is the largest carry from one column to the next?
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2. Find the largest possible carry for 5, 6, and 7 addends. Describe any pattern you notice.
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3. Does your pattern continue for more addends? If not, propose a corrected pattern.
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4. How would this carry behavior change in other bases (e.g., base twelve)? Explain.
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12.

1.

1. \(\displaystyle 158_{ten} - 146_{ten}\)
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2. \(\displaystyle 634_{ten} - 237_{ten}\)
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3. \(\displaystyle 745_{eight} - 367_{eight}\)
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4. \(\displaystyle B45_{twelve} - 367_{twelve}\)
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5. \(\displaystyle 311_{four} - 302_{four}\)
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2.

1. \(\displaystyle 431_{ten} - 141_{ten}\)
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2. \(\displaystyle 416_{ten} - 235_{ten}\)
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3. \(\displaystyle 331_{four} - 123_{four}\)
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4. \(\displaystyle B72_{twelve} - 801_{twelve}\)
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5. \(\displaystyle 341_{eight} - 77_{eight}\)
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3.

1. \(\displaystyle 449_{ten} - 277_{ten}\)
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2. \(\displaystyle 993_{ten} - 685_{ten}\)
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3. \(\displaystyle 860_{twelve} - 7B8_{twelve}\)
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4. \(\displaystyle 320_{four} - 133_{four}\)
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5. \(\displaystyle 567_{eight} - 373_{eight}\)
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4.

1. \(\displaystyle 934_{ten} - 374_{ten}\)
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2. \(\displaystyle 747_{ten} - 159_{ten}\)
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3. \(\displaystyle 100_{four} - 22_{four}\)
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4. \(\displaystyle 8A9_{twelve} - 7BB_{twelve}\)
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5. \(\displaystyle 751_{eight} - 27_{eight}\)
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5.

1. \(\displaystyle 1222_{ten} - 237_{ten}\)
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2. \(\displaystyle 4370_{twelve} - BBB_{twelve}\)
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3. \(\displaystyle 707_{eight} - 561_{eight}\)
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6.

1. \(508_{ten} - 479_{ten}\)

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   \begin{gather*} \phantom{-}\;{}^{4}\!5\;\;{}^{1}\!0\;\;{}^{1}\!8\\ - \;\;\;\;4\;\;\;\;7\;\;\;\;9\\ \phantom{-}\;\;\;\;\;\;\;3\;\;\;\;9 \end{gather*}

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2. \(546_{ten} - 309_{ten}\)

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   \begin{gather*} \phantom{-}\;5\;\;{}^{1}\!4\;\;{}^{1}\!6\\ - \;{}^{4}\!3\;\;{}^{10}\!0\;\;9\\ \phantom{-}\;\;1\;\;\;\;4\;\;\;\;7 \end{gather*}

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3. \(781_{twelve} - 434_{twelve}\)

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   \begin{gather*} \phantom{-}\;7\;\;8\;\;{}^{1}\!1\\ \phantom{-}\;4\;\;{}^{4}\!3\;\;4\\ \phantom{-}\;\;3\;\;\;4\;\;\;9 \end{gather*}

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4. \(639_{ten} - 42_{ten}\)

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   \begin{gather*} \phantom{-}\;6\;\;{}^{1}\!3\;\;9\\ - \;\;\;\;\;4\;\;\;2\\ \phantom{-}\;6\;\;\;9\;\;\;7 \end{gather*}

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7.

1. Compute \(567 - 456\text{.}\)
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2. Compute \(456 - 567\) on a calculator. What do you notice? How could an algorithm produce the same result?
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3. Let \(a,b \in \mathbb{N}\) with \(a > b\text{.}\) Express \(b - a\) in terms of \(a - b\) and explain why this helps.
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4. Write clear steps for handling cases with a negative result, using your preferred subtraction algorithm.
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8.

1. If revenue is \(287804_{twelve}\text{,}\) compute the profit (in base twelve). Show how you combined expenses from different bases.
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2. Next year, expenses stay the same but revenue is \(100000_{ten}\text{.}\) Dave claims the profit is \(A9014\) (base twelve). Find the actual profit (state the base) and explain Dave’s error.
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1.

1. \(\displaystyle 10^8 \times 10^3\)
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2. \(\displaystyle 8^{17} \times 8^9\)
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3. \(\displaystyle 7^9 \times 7^7 \times 7^3\)
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2.

1. \(\displaystyle 300 \times 700\)
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2. \(\displaystyle 9 \times 3000\)
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3. \(\displaystyle 3400 \times 200\)
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4. \(\displaystyle 400 \times 20 \times 30\)
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3.

1. \(\displaystyle 33 \times 68\)
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2. \(\displaystyle 450 \times 387\)
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3. \(\displaystyle 9500 \times 4567\)
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4.

1. \(\displaystyle 45 \times 80\)
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2. \(\displaystyle 608 \times 789\)
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3. \(\displaystyle 3249 \times 5213\)
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4. \(\displaystyle 399 \times 457 \times 68\)
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5.

1. \(\displaystyle 11 \times 47\)
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2. \(\displaystyle 730 \times 888\)
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3. \(\displaystyle 16 \times 543\)
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4. \(32 \times 64 \times 435\) (hint: group powers of two)
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6.

7.

8.

9.

10.

1. \(\displaystyle 14_{twelve} \times 89_{twelve}\)
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2. \(\displaystyle 373_{eight} \times 762_{eight}\)
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3. \(\displaystyle 222_{four} \times 231_{four}\)
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11.

1. Top row shows \(831\text{.}\) The three partial products written beneath are \(600\text{,}\) \(18{,}000\text{,}\) and \(4{,}800{,}000\) (shifted by place). What was the other factor?
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2. Top row shows \(456\text{.}\) The partial products include \(18\text{,}\) \(150\text{,}\) \(1200\text{,}\) \(120\text{,}\) \(1000\text{,}\) \(8000\text{,}\) \(60\text{,}\) \(5000\text{,}\) \(40{,}000\) (each appropriately shifted). What was the other factor?
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3. Top row shows \(139\text{.}\) The partial products include \(18\text{,}\) \(60\text{,}\) \(200\text{,}\) \(630\text{,}\) \(2100\text{,}\) \(7000\text{,}\) \(6300\text{,}\) \(21{,}000\text{,}\) \(70{,}000\text{.}\) What was the other factor?
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12.

1. For \(954 \times 211\text{,}\) a student lists nine one-digit lines (4, 5, 9, 4, 5, 9, 8, 10, 18) and sums to get \(72\text{.}\) What place-value mistake is being made?
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2. For \(761 \times 990\text{,}\) a student treats \(990\) as \(99\) and forgets the trailing zero shift. Explain how place shifting should work and what the correct structure of partial products is.
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3. For \(750 \times 878\text{,}\) the partial products mishandle multiples of \(10\) and \(40\text{.}\) Explain how to organize the computation so every tens/hundreds place is accounted for correctly.
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4. For \(634 \times 272\text{,}\) a neatly organized set of shifted partial products leads to the correct total. State why this layout is valid (identify the properties used).
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1.

1. \(\displaystyle 402 \div 6\)
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2. \(\displaystyle 783 \div 9\)
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3. \(\displaystyle 565 \div 7\)
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4. \(\displaystyle 888 \div 10\)
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2.

1. \(\displaystyle 755 \div 5\)
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2. \(\displaystyle 988 \div 3\)
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3. \(\displaystyle 441 \div 4\)
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4. \(\displaystyle 653 \div 7\)
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3.

1. \(\displaystyle 525 \div 15\)
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2. \(\displaystyle 481 \div 13\)
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3. \(\displaystyle 1836 \div 25\)
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4. \(\displaystyle 1670 \div 11\)
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5. \(\displaystyle 67894 \div 8\)
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4.

1. \(\displaystyle 11113_{four} \div 13_{four}\)
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2. \(\displaystyle 10123_{four} \div 22_{four}\)
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3. \(\displaystyle 1524_{eight} \div 6_{eight}\)
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4. \(\displaystyle 21224_{eight} \div 11_{eight}\)
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5. \(\displaystyle 4A56_{twelve} \div 6_{twelve}\)
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6. \(\displaystyle BB67_{twelve} \div 10_{twelve}\)
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5.

1. How many whole cookies does each friend get, and how many are left over?
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2. Consider instead sharing between two people. Decide if this is possible without carrying out the full division, and explain your test.
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3. Is sharing \(501\) cookies among three people possible without division? Explain your reasoning clearly (hint: examine a quick test using the digits).
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6.

1. \(\displaystyle (1232 \div 8) + 776\)
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2. \(\displaystyle (6399 \div 9) - 254\)
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3. \(\displaystyle (1778 \div 7) - 2344\)
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4. \(\displaystyle (1040 \div 13) + (345 \times 7670)\)
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5. \(\displaystyle (12345 \div 15) - (565 \times 887)\)
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1.

1. State precisely what a “carry” represents in base \(b\text{,}\) and why its value is always an integer between \(0\) and \(b-1\text{.}\)
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2. Justify that adding the units column and writing the remainder “mod \(b\)” with a carry of \(\left\lfloor \dfrac{\text{column sum}}{b}\right\rfloor\) preserves the total value of the sum.
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3. Connect the algorithm to expanded form (powers of \(b\)) showing equality of values before and after carrying.
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2.

3.

1. Prove algebraically that the difference is unchanged.
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2. Give a base-ten number line model and a base-eight blocks model to illustrate the invariant.
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3. Discuss when same-change is especially helpful (e.g., making the subtrahend a round number such as \(1000,\, 100_{eight},\, 1{,}00_{twelve}\)).
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4.

1. Explain why adding \(b\) to one column of both numbers preserves the overall difference.
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2. Describe the relationship between equal additions in base ten and the \(9\)’s complement method; generalize to base \(b\) and \((b-1)\)’s complements.
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3. Give a worked example in base twelve and narrate each invariant you are using.
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5.

1. Using an area diagram, decompose \((a_1b + a_0)\times (c_1b + c_0)\) and write the sum of four partial products.
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2. Explain how the shifts in the standard algorithm correspond to multiplying by powers of \(b\text{.}\)
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3. Why does the order of writing partial products not affect the final sum? Name the properties used.
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6.

1. Identify where each digit–digit product appears in the lattice and which place value it contributes to.
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2. Explain the diagonal carries as “collecting equal powers of \(b\text{.}\)”
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7.

1. Explain why \(a\times b = \sum 2^k \cdot (\text{odd-part contribution})\) corresponds to the binary expansion of one factor.
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2. Prove correctness by induction on the number of halving steps, or by writing \(b\) in base two.
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3. Give an example where this method is faster than the standard algorithm and an example where it is slower, and explain why.
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8.

1. State what each quotient digit represents and why choosing the largest digit that “fits” is a greedy but correct step.
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2. Explain why the remainder at each stage is strictly smaller than the divisor (in base \(b\) units), ensuring termination.
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3. Relate the algorithm to writing the dividend as \(q\cdot d + r\) with \(0\le r<d\) at each truncated place.
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9.

1. Explain how dividing by \(25\) can be transformed into multiplying by \(4\) and dividing by \(100\text{,}\) and why place value makes this efficient.
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2. Give an example where factoring the divisor into primes yields a faster mental computation than straight long division.
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10.

1. Give the maximum possible carry from a single column when adding \(k\) two-digit base-\(b\) numbers (justify your bound in terms of \(b\) and \(k\)).
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2. Describe a subtraction example in base eight where equal additions substantially reduces the number of borrows compared to the standard method; explain why.
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11.

1. Multiply \(250\times 48\text{.}\)
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2. Subtract \(10{,}000 - 4{,}997\text{.}\)
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3. Divide \(1{,}836\div 25\text{.}\)
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4. Multiply \(8A9_{twelve}\times BBB_{twelve}\text{.}\)
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12.

1. Student A multiplies \(340\times 206\) and writes partial products \(6{,}80{,}4\) (digitwise) with no shifting. Identify the misconception and rewrite the computation correctly, explaining the role of place-value shifts.
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2. Student B computes \(7003 - 5687\) by “same-change” as \(7000 - 5684\) and then subtracts to get \(316\text{.}\) Is the method valid? If so, state the invariant and fix any arithmetic mistakes; if not, explain the flaw.
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13.

1. In any base \(b\ge 2\text{,}\) lattice multiplication is just the distributive law applied to expanded forms.
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2. For integers, the remainder after division by \(d\ne 0\) is unique with \(0\le r<|d|\text{.}\)
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3. Partial sums and partial differences are both instances of associativity (grouping by like place values).
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14.

1. Give an upper and lower bound for \(608\times 789\) using one-digit rounding, and explain how such bounds can reveal place-shift errors.
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2. Give a bound check for \(67894\div 8\) that would flag a misplaced quotient digit.
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15.