Final answer:
The Canadian Prime Minister from the 1870s who shared the same name as the Scottish explorer is Sir Alexander Mackenzie. Sir John A. Macdonald was the first Prime Minister, but the explorer Alexander Mackenzie was known for his attempts to find the Northwest Passage.
Step-by-step explanation:
The Canadian Prime Minister from the 1870s who shared the same name as the Scottish explorer who mapped the longest river within Canada is Sir Alexander Mackenzie. The explorer, Alexander Mackenzie, was an officer of the British North West Company and undertook significant expeditions, including reaching the Pacific Ocean. He was in search of the elusive Northwest Passage, a quest that many Europeans and later Americans pursued.
In contrast, the first Prime Minister of Canada was Sir John A. Macdonald, whose tenure started in the 1860s. He was instrumental in purchasing the Northwest Territories and in adding several provinces to the Dominion of Canada, such as Manitoba, British Columbia, and Prince Edward Island. He also played a critical role in the completion of the Canadian Pacific Railroad in 1885.
Therefore, despite their similar sounding names, Sir John A. Macdonald and the explorer Alexander Mackenzie were indeed different individuals associated with Canada's early development and expansion. The correct answer is (c) Sir Alexander Mackenzie.