Grade

Type of questions

Mathematics Senior High

⑶で最後のpの倍数の個数を求める式がよくわかりません。

例題260 互いに素な自然数の個数 を自然数とする.m≦nでmとnが互いに素である自然数mの個数 をf(n) とするとき、 次の問いに答えよ. (1) f (15) を求めよ. (2) f (pg) を求めよ.ただし, p, g は異なる素数とする. (3) f(p) を求めよ.ただし、pは素数, kは自然数とする. (名古屋大・改) 考え方 (1) 「m≦nでmとnが互いに素である自然数mの個数をf(n) とする」とはどう いうことかを(1) の f (15) をもとにして考えてみる. f(15) はn=15 の場合であるから, ☆「m≦15 でmと15が互いに素である自然数mの個数は (15) となる。 つまり, (1)を言い換えると次のようになる. 合 (1) 15=3.5 であるから, 15と互いに素でない自然数, すなわち, 3の倍数または5の倍数であり, 15以下の 自然数は, 3,6,9,12, 15,510の7個である. よって, 15 と互いに素な自然数の個数は, f(15)=15-7=8 もつやっ魂 (2) gは異なる素数であるから、 pg と互いに素でな い自然数, すなわち, pの倍数またはgの倍数であり, 以下の自然数は, ①の倍数 10 2.⑩..... (q-1)0, HTA 教えた 「15 以下の自然数で15と互いに素である自然数はいくつあるか」 (2)(1)では,15=3・5 であった.(2)ではggは互いに素より(1)と同様にして 考えてみる. 個 ⑨の倍数 1⑨ 2.⑦ .…... (p-1) @カ@のか個 が互いに 3Mの数) ⑩9の倍数 1 SCAND り (q+p-1) 1 よって, bg と互いに素な自然数の個数は 1.2.3.....pa f(pq)=pa(g+p-1) Focus の 個 P9以下の自然数の **** = pg-p-g+1=(-1)(g-1) (3) p, kは自然数であるから, が以下の自然数は CHA (1.2.3.....PR) 個ある. pは素数であるから,以下の自然数の倍数 は全部で, pp=1個) 123 したがって, f(p")=pk-pk-1 練習 260 (g)とする. *** 「互いに素である」の 否定 「互いに素でな い」 を考える. 5 (1) を一般的に考える. p=3,g=5 としてみ ると見通しがよくなる. pg÷p=g(個) pg÷g=p(個) (1) f(77) を求めよ. (2) f (pg) = 24 となる p, g の組をすべて求め上 pg 以下の自然数 の倍数 STY 互いに素である自然数の個数は、補集合の考えを利用せよ ☆互いに素でない(1以外に共通の縞ある)もの数える 9の倍数 P9の倍数 (p.185 例題 94 参照) f(n) をオイラー関数 という. (p.538 Column 参照) ががが(-1) 例題260 の f (n) について次の問いに答えよ. ただし, p, g は異なる素数 改) 12 女 (c た C

Waiting for Answers Answers: 0
English Senior High

答えに解説がなくて困ってます。 下の長文を翻訳してください。

〔Ⅰ〕 次の英文を読み. 設問 1~21 に答えよ。 Sandy lives in an apartment so small that when she comes home from shopping, she has to decide what to move out to make room for her purchases. She struggles day-to-day to feed and clothe herself and her four-year-old daughter on money from freelance writing jobs and helping neighbors. (2) Her ex-husband has long since disappeared down some unknown highway, probably never to be heard from again. As often as not, her car decides it needs a day off and refuses to start. That means bicycling (weather permitting), walking or asking friends for a ride. 13 The things most Americans consider essential for survival- a television. microwave, big freezer and high-priced sneakers are far down Sandy's list of "maybe someday" items. (5) Nutritious food, warm clothing, an affordable apartment, student loan payments, books for her daughter, absolutely necessary medical care and an occasional movie eat up what little money there is to go around. Sandy has knocked ) more doors than she can recall, trying to find (7) a decent job, but there is always something that doesn't quite fit-too little experience or not the right kind, or hours that make child care impossible. Sandy's story is not unusual. Many single parents and older people struggle with our economic structure, falling into the gap between being truly self-sufficient and being poor enough that the government will provide assistance. What makes Sandy unusual is her outlook. "I don't have much in the way of stuff or the American dream," she told me with a genuine smile. "Does that bother you?" I asked. "Sometimes. When I see another little girl around my daughter's age who has nice clothes and toys, or who is riding around in a fancy car or living in a fine house, then I feel bad. Everyone wants to do well for their children." she replied. "But you're not angry?" "What's to be angry (9) and I have what is really important in life," she replied. "And what is that?" I asked. (10) "As I see it, no matter how much stuff you buy, no matter how much )? We aren't starving or freezing to death. (11) money you make. you really only get to keep three things in life." she said. "What do you mean by 'keep?" (12) "I mean that nobody can take these things away from you." "And what are these three things?" I asked. "One, your experiences: two, your true friends; and three, what you grow inside yourself." she told me without hesitation. (13) For Sandy, "experiences" don't come on a grand scale. They are so-called ordinary moments with her daughter, walks in the woods, napping under a shady tree, listening to music, taking a warm bath or baking bread. Her definition of friends is more expansive. "True friends are the ones (15) who never leave your heart, even if they leave your life for a while. Even after years apart. you pick up with them right where you left off, and even if they die, they're never dead in your heart," she explained. 16 ) to each of us. (17 As for what we grow inside, Sandy said, "That's ( isn't it? I don't grow anger or sorrow. I could if I wanted to, but I'd rather not." "So what do you grow?" I asked. Sandy looked warmly at her daughter and then back to me. She pointed toward her own eyes, which were shining with tenderness. gratitude and a sparkling joy. "I grow this." From the book Chicken Soup for the Woman's Soul by Jack Canfield. Mark Victor Hansen. Jennifer Read Hawthorne, and Marci Shimoff. Copyright 2012 by Chicken Soup for the Soul Publishing, LLC. Published by Backlist. LLC. a unit of Chicken Soup for the Soul Publishing. LLC. Chicken Soup for the Soul is a registered trademark of Chicken Soup for the Soul Publishing, LLC. Reprinted by permission. All rights reserved.

Unresolved Answers: 1