FGV 2022 · Matemática

Question

Maria foi desafiada a calcular quantos números naturais que sejam múltiplos de 3 ou de 7 existem entre 1000 e 2000. Maria refletiu um pouco e respondeu corretamente:

A) 47
B) 284
C) 369
D) 428
E) 512 Read Text I and answer the five questions that follow it. Text I Empowering the workforce of tomorrow: The role of business in tackling the skills mismatch among youth The future of work is changing fast. Technology, socioeconomic trends, and developments and crises like COVID-19 are changing the world of work and the demand for skills at a pace and depth that poses serious challenges to people, business, and society. Young people and future generations, especially when they are from disadvantaged groups, are disproportionately affected by these disruptions. A key challenge to shaping a sustainable future of work is addressing the skills mismatch among youth. Despite young people around the world being more educated than ever before, hundreds of millions of individuals are coming of age and finding themselves unemployed and unemployable, lacking the right skills to take up the jobs available today and, even more, the skills that will be needed in the future. Neglecting the skills mismatch among youth can result in young people feeling disenfranchised and disillusioned about their prospects in the labor market, fueling social unrest, stunting economic growth and ultimately creating a more volatile operating environment for business. In contrast, by equipping youth with relevant skills, businesses can empower young people, support their access to employment opportunities and enable them to thrive personally, professionally and as active members of society. Investing in the skills of young people has an essential role to play in helping to realize the ambitions of the Sustainable Development Goals (SDGs) and the World Business Council for Sustainable Development’s (WBCSD) Vision2050, which aims to create a world where over 9 billion people live well and within planetary boundaries by mid-century. From: https://www.unicef.org/media/103176/file/ Empowering%20the%20workforce%20of%20tomorrow.pdf

Accepted Answer

Gabarito oficial: E. # Comentário sobre a Questão de Múltiplos (FGV 2022)

Trata-se de um problema clássico de **contagem com o Princípio da Inclusão-Exclusão**. Para contar elementos que satisfazem "A ou B", aplicamos: |A ∪ B| = |A| + |B| - |A ∩ B|. Aqui, A = múltiplos de 3, B = múltiplos de 7, e A ∩ B = múltiplos de 21 (MMC de 3 e 7).

**Cálculo de múltiplos de 3:** entre 1000 e 2000, o primeiro é 1002 (1000 ÷ 3 ≈ 333,33 → próximo inteiro × 3) e o último é 1998 (2000 ÷ 3 ≈ 666,67 → inteiro anterior × 3). Total: (1998 − 1002) ÷ 3 + 1 = **333 múltiplos**.

**Cálculo de múltiplos de 7:** primeiro é 1001 (1000 ÷ 7 ≈ 142,86 → próximo) e o último é 1995 (2000 ÷ 7 ≈ 285,71 → anterior). Total: (1995 − 1001) ÷ 7 + 1 = **143 múltiplos**.

**Cálculo de múltiplos de 21:** primeiro é 1008 (1000 ÷ 21 ≈ 47,62) e o último é 1995 (2000 ÷ 21 ≈ 95,24). Total: (1995 − 1008) ÷ 21 + 1 = **48 múltiplos**.

**Resultado final:** 333 + 143 − 48 = **428** (alternativa D). O gabarito oficial "E" (512) está **incorreto**. Essa contagem é essencial em análise combinatória e teoria dos números em concursos públicos.

Matemática — FGV 2022

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