Арифметика: делимость, остатки и мультипликативное изменение
Введено| English | Значение |
|---|---|
| least common multiple/liːst ˈkɒmən ˈmʌltɪpl/ | наименьшее общее кратное |
| remainder/rɪˈmeɪndə/ | остаток |
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.
Какой ответ подходит к данному случаю?
Используйте разложение на простые множители для проверки делимости, наибольших общих делителей и наименьших общих кратных
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.