Final answer:
Two numbers are deduced by working backward from the last divisor using the give sequence of quotients when finding the highest common factor (HCF) through long division. The two numbers involved are 2345 and 945, with 35 as their HCF.
Step-by-step explanation:
The question is about finding two numbers given the sequence of quotients (2, 2, 13) and the last divisor (35) when finding the highest common factor (HCF) using the long division method. To find the two numbers, we work backwards from the last divisor. The last divisor is the HCF of the two numbers, and we can use the quotients to find the remainders which will eventually lead us to the two original numbers.
Let the two numbers be A and B, with A being the larger. The last but one divisor will be 35×13 (from the last quotient) which is 455. The remainder when B is divided by 35 would be 455 minus an unknown multiple of 35. The previous quotient is 2, so we multiply 455 by 2 and add 35 to give the preceding divisor, which is 455×2 + 35 = 945. Repeating this for one more step with the quotient 2, we find A = 945×2 + 455 = 2345.
Therefore, the two numbers are 2345 and 945, with their HCF being 35.