חישוב: חלוקיות, שאריות ושינוי מכפילי
הוצג| English | משמעות |
|---|---|
| least common multiple/liːst ˈkɒmən ˈmʌltɪpl/ | מכפלה משותפת קטנה ביותר |
| remainder/rɪˈmeɪndə/ | שארית |
A decision before an answer
- A 20% decrease followed by a 20% increase does not restore the starting value: the second percentage uses a smaller base.
- Your goal: Use prime factorisation for divisibility, greatest common factors and least common multiples.
Read the relationship
- An integer divisible by d can be written dk; a remainder r gives dk+r with 0≤r<d for a positive divisor. Prime factorisation identifies divisibility and common multiples. The greatest common factor takes shared prime powers at their smaller exponents; the least common multiple takes every needed prime power at its larger exponent.
- Interpret remainders with the original divisor.
LCM of 12 and 18 is:
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12=2²·3 and 18=2·3²; maximum powers give 36.
Use the defining rule
- Keep odd/even and positive/negative properties separate. A product of two odd integers is odd; a product with an even integer is even. A statement about integers need not hold for real numbers. For quantitative comparison, exploit a remainder form to expose possible cases rather than assume the smallest positive example is the only value.
- Combine ratios and percentage changes with the correct reference total.
A price 200 rises 10% then falls 10%. Final price:
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200·1.1·0.9=198.
Check the conditions
- A ratio a:b allocates a whole into a+b equal parts when the groups exhaust it. A percentage change multiplies the original value by 1+r or 1−r with r expressed as a decimal. Successive changes multiply factors. A part-of-a-part calculation changes the reference group at each stage; label the denominator before multiplying.
- Combine ratios and percentage changes with the correct reference total.
Known: $18=2\cdot3^2$ and $24=2^3\cdot3$. Shared smaller powers give GCF six; needed larger powers give LCM seventy-two. If $n=5k+2$, then $2n=5(2k)+4$, so the remainder on division by five is four.
For a model price of 100 CNY decreased then increased by 20%, multiply the two changes:
Dividing 250 in the ratio 2:3 gives 100 and 150: each of five parts is fifty.
If n has remainder 3 on division by 7, n+2 has remainder ____.
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n=7k+3, so n+2=7k+5.
Apply the task format
- Estimate magnitude and check integer restrictions after solving. Fractions, roots and negative exponents follow their ordinary domain restrictions; even roots represent the nonnegative principal value. Do not cancel a term in a sum as if it were a factor. Exact arithmetic can be faster and safer than decimal calculator entries.
- Combine ratios and percentage changes with the correct reference total.
Equal percentage increases and decreases use different bases. A remainder must be reported relative to the named divisor.
איזו תשובה מתאימה למקרה זה?
שימוש בפירוק לגורמים ראשוניים לבדיקת חלוקה, מחלקים משותפים מרובים ומכפלים משותפים קטנים
Every real number has integer parity.
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Odd/even applies to integers, not arbitrary real numbers.
Keep the distinctions
- least common multiple 最小公倍数 — The smallest positive integer divisible by each given positive integer.
- remainder 余数 — The nonnegative amount left after division by a positive integer, smaller than that divisor.
- Use prime factorisation for divisibility, greatest common factors and least common multiples.
- Interpret remainders with the original divisor.
- Combine ratios and percentage changes with the correct reference total.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.