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The most likely diagnosis for the 64-year-old woman presenting with fatigue, sluggishness, and weight gain is hypothyroidism. This condition occurs when the thyroid gland does not produce enough thyroid hormones, which are crucial for regulating metabolism. The symptoms she is experiencing align with common manifestations of hypothyroidism, including fatigue, a sense of sluggishness, and unexplained weight gain due to a slowed metabolic rate.

In contrast, hyperthyroidism, characterized by excessive production of thyroid hormones, would typically present with symptoms such as weight loss, increased energy levels, and restlessness—symptoms that are inconsistent with the woman's complaints. Hyperparathyroidism is associated with elevated calcium levels and may lead to symptoms such as kidney stones, fatigue, and muscle weakness, which differ from those described in the case. Likewise, hypoparathyroidism involves low levels of parathyroid hormone, leading to low calcium levels and symptoms like muscle cramps and tingling, rather than the signs experienced by the woman in question.

Thus, the combination of her age and presenting symptoms strongly supports a diagnosis of hypothyroidism as the most likely explanation for her condition.