Question

1. write an expression that can be used to check the quotient of 646 ÷ 3.

Explanation:

Step1: Recall division check formula

In division, dividend = quotient × divisor + remainder. For \(646\div3\), let quotient be \(q\), divisor \(3\), dividend \(646\). The check expression is \(646 = q\times3 + r\) (but if we assume we found quotient \(q\) and remainder \(r\), to check, we can use \(q\times3 + r = 646\). If we just consider the quotient (ignoring remainder for the expression structure), the basic check is quotient × divisor should relate to dividend. So the expression to check the quotient (let \(q\) be quotient) is \(q\times3\) (and then add remainder if needed, but the problem says "check the quotient", so the expression using quotient, divisor, and dividend relation is \(q\times3 + r = 646\), but if we do the division \(646\div3 = 215\) with remainder \(1\), so the check expression is \(215\times3 + 1 = 646\), but as a general expression for the quotient \(q\) of \(646\div3\), the check expression is \(q\times3 + r = 646\), or if we write it as \(q\times3=\) (dividend - remainder), but the standard way to check division is quotient × divisor + remainder = dividend. So the expression to check the quotient (after finding it) is \(q\times3 + r = 646\), where \(q\) is the quotient and \(r\) is the remainder. But if we just need the expression using the quotient, divisor, and the relation, it's \(q\times3\) (and then adjust for remainder). But the problem is to write an expression to check the quotient, so the correct expression is quotient × 3 + remainder = 646. If we perform \(646\div3\), we get quotient \(215\) and remainder \(1\), so the expression is \(215\times3 + 1\) (which should equal 646). But as a general expression for the quotient \(q\) of \(646\div3\), the check expression is \(q\times3 + r = 646\), or more simply, if we let \(q\) be the quotient, the expression to check is \(q\times3\) (and then see if adding remainder gives dividend). But the standard formula for checking division is \( \text{quotient} \times \text{divisor} + \text{remainder} = \text{dividend} \). So for \(646\div3\), the expression to check the quotient (let \(q\) be quotient, \(r\) remainder) is \(q \times 3 + r = 646\).

Step2: Form the expression

After dividing \(646\) by \(3\), we get quotient \(q = 215\) and remainder \(r = 1\). So the expression to check is \(215\times3 + 1\) (which equals \(646\)). But as a general expression for the quotient of \(646\div3\), the check expression is \( \text{quotient} \times 3 + \text{remainder} = 646 \). If we write it with the quotient found, it's \(215\times3 + 1\) (or as a variable expression, \(q\times3 + r = 646\) where \(q\) is the quotient of \(646\div3\) and \(r\) is the remainder).

Answer:

The expression to check the quotient of \(646\div3\) (letting \(q\) be the quotient and \(r\) be the remainder) is \(q\times3 + r = 646\). If we compute the quotient (\(215\)) and remainder (\(1\)) for \(646\div3\), the specific check expression is \(215\times3 + 1\) (which equals \(646\)). So the expression is \(\boldsymbol{215\times3 + 1}\) (or in variable form \(q\times3 + r = 646\) where \(q\) is the quotient of \(646\div3\)).