answer:1. Time for personal growth and self-discovery - Taking a gap year provides an opportunity to step out of the typical academic or professional routine and allows individuals to explore new interests and gain a greater sense of self-awareness. This can lead to increased confidence, improved decision-making skills, and a clearer sense of direction in life. 2. Enhanced cultural awareness and global perspective - Traveling during a gap year exposes individuals to diverse cultures and ways of life. This experience can foster empathy, tolerance, and a greater understanding of different worldviews. These skills are invaluable in our increasingly globalized society and can also boost employability in certain fields. 3. Improved academic and career outcomes - Many students who take a gap year return to their studies with renewed motivation and focus. They often perform better academically and have a clearer sense of their goals and career path. Additionally, gap year experiences such as volunteer work, internships, or language immersion programs can enhance resumes and set individuals apart in the job market.

answer:1. We start by investigating the number ( p ): [ p = underbrace{11 cdots 11}_{n text{ digits}} left(10^{3n} + 9 times 10^{2n} + 8 times 10^n + 7 right) ] where (underbrace{11 cdots 11}_{n text{ digits}}) is an integer consisting of ( n ) repetitions of the digit 1. 2. Given that (underbrace{11 cdots 11}_{n text{ digits}}) is divisible by 1987, we conclude that ( p ) is divisible by 1987 due to the factor multiplication in the expression. 3. Next, we consider the number ( q ): [ q = underbrace{11 cdots 11}_{(n+1) text{ digits}} left(10^{3(n+1)} + 9 times 10^{2(n+1)} + 8 times 10^{n+1} + 7 right) ] where (underbrace{11 cdots 11}_{(n+1) text{ digits}}) is an integer consisting of ( n+1 ) repetitions of the digit 1. 4. We know from modular arithmetic properties that: [ 10^n equiv 9 cdot underbrace{11 cdots 11}_{n text{ digits}} + 1 mod 1987 ] 5. Therefore, we can express the powers of ( 10 ) involved in ( q ) as follows: [ begin{array}{l} 10^{3(n+1)} = left(10^nright)^3 times 10^3, 10^{2(n+1)} = left(10^nright)^2 times 10^2, 10^{n+1} = 10^n times 10, end{array} ] 6. From the above, we note the following congruences: [ begin{array}{l} 10^{3(n+1)} equiv 1000 mod 1987, 10^{2(n+1)} equiv 100 mod 1987, 10^{n+1} equiv 10 mod 1987. end{array} ] Each of these is divisible by ( underbrace{11 cdots 11}_{n text{ digits}} ). 7. Consider ( q ): [ q = underbrace{11 cdots 11}_{(n+1) text{ digits}} left(10^{3(n+1)} + 9 times 10^{2(n+1)} + 8 times 10^{n+1} + 7right), ] and since each term inside the parentheses is congruent to a multiple of ( underbrace{11 cdots 11}_{n text{ digits}} ) 8. We conclude that ( q ) can also be divisible by 1987 since the multiplier includes a sequence that sums to a number having base congruent modulo 1987. # Conclusion: [ boxed{p text{ and } q text{ are both divisible by 1987}} ]

answer:The best vitamins to take vary depending on factors such as age, sex, diet, and health status. However, here are some essential vitamins that are commonly recommended: A well-balanced diet usually provides adequate amounts of vitamins, but some people may require supplements to fill nutritional gaps. Vitamin D is crucial for bone health, immune function, and mood regulation, and many people are deficient due to limited sun exposure or inadequate dietary intake. Omega-3 fatty acids support heart health and brain function, and can be obtained through fatty fish consumption or supplements. Calcium is essential for bone health, particularly for older adults, pregnant women, and individuals with a high risk of osteoporosis. Vitamin B12 plays a vital role in energy production, nerve function, and the formation of red blood cells, and deficiencies are common among vegans and older adults. Other important vitamins and nutrients include vitamin C for immune function, vitamin E for antioxidant properties, and iron for healthy red blood cells. It's essential to consult with a healthcare professional before starting any supplements to determine the best vitamins and dosages for individual needs. They can help identify potential deficiencies and recommend suitable supplements to support overall health and well-being.

answer:Taking excessive amounts of certain vitamins can be toxic and cause adverse health effects. Fat-soluble vitamins, in particular, can be toxic in high doses since they accumulate in the body and are not easily excreted. Vitamin A toxicity can occur with high doses, typically above 10,000 IU per day, causing symptoms such as headaches, dizziness, and liver damage. Prolonged excessive intake can lead to more severe health issues, including bone fractures and birth defects. Vitamin D toxicity is also possible, especially with doses exceeding 4,000 IU per day. Elevated levels of vitamin D can cause hypercalcemia, a condition characterized by high calcium levels in the blood, leading to symptoms like nausea, vomiting, and kidney damage. High doses of vitamin E, typically above 1,000 IU per day, can increase the risk of bleeding due to its anticoagulant effects. Vitamin K, while essential for blood clotting, can also be toxic in excessive amounts, particularly for people taking blood thinners. In contrast, water-soluble vitamins like vitamins B and C are generally less likely to cause toxicity since excess amounts are excreted in the urine. However, high doses of certain B vitamins, such as vitamin B6, can still cause adverse effects like nerve damage. It is essential to consult with a healthcare professional before taking any supplements to ensure safe and effective use. They can help determine the optimal dosage and monitor for potential toxicity.