Final answer:
Patient A most likely has hypothyroidism and Patient B most likely has hyperthyroidism.
Step-by-step explanation:
Based on the symptoms described, Patient A is most likely to have hypothyroidism. Hypothyroidism is characterized by weight gain, cold sensitivity, low heart rate, and fatigue. These symptoms are consistent with an underactive thyroid gland. Patient B, on the other hand, is most likely to have hyperthyroidism. Hyperthyroidism is characterized by weight loss, profuse sweating, increased heart rate, and difficulty sleeping. These symptoms are consistent with an overactive thyroid gland.