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When you're dividing numbers, sometimes things don't divide perfectly. This situation leaves us with a remainder. The remainder represents the 'leftover' part that couldn’t be evenly divided.

Imagine you're sharing 167 candies among 3 friends. You give each friend 55 candies. Then, you have 2 candies left over because 167 doesn’t divide evenly by 3.

In a mathematical division, the remainder can be expressed like this: if given a dividend (167) and a divisor (3), you can say:

• Dividend = Quotient × Divisor + Remainder
• For this problem, it looks like: 167 = 55 × 3 + 2

Recognizing the remainder helps understand how complete the division is, and it's crucial for ensuring calculations are accurate.

The quotient is the key result we obtain from dividing two numbers. In our exercise, when we divided 167 by 3, the quotient was 55.

How do we know this? We figured out how many times 3 can fit into 167. First, we checked 16 (the first two digits of 167). Since 3 fits into 16 five times (as 3 times 5 is 15), we moved forward.

Continuing, we brought down the next digit, 7, resulting in 17. Again, 3 fits into 17 five times, giving us another quotient digit.

Thus, the full quotient of 167 divided by 3 is 55. This quotient tells us that 167 contains 55 groups of 3, with a small amount remainder." Recognizing the quotient in division is essential for understanding the problem's solution, and it's used in re-checking the division through multiplication.

After working through a division problem, you may wonder if you've done it right. That’s where multiplicative verification comes in.

In this exercise, you can ensure the division is correct by reversing it through multiplication. Take the quotient (55) and multiply it by the divisor (3).

• Calculate: \( 55 \times 3 = 165 \)

Then, add the remainder (2):

• \( 165 + 2 = 167 \)

If your sum equals the dividend (167), the division is accurate.

Multiplication check is a reliable and handy tool not just to verify division, but also to reinforce your understanding of fundamental arithmetic operations.