A third-grade teacher displays a multiplication chart. The teacher highlights the multiples of 6 and asks the students to identify patterns they see in the numbers that are highlighted. Which THREE of the following are correct student responses?
The sums of the digits are all multiples of 3.
The multiples of 6 exhibit a unique property where the sum of their digits is always a multiple of 3. This is due to the fact that any number that is a multiple of 6 must also be divisible by both 2 and 3, and the divisibility rule for 3 states that a number is divisible by 3 if the sum of its digits is divisible by 3.
This statement is true as it reflects the mathematical property of multiples of 6. Since all multiples of 6 are also multiples of 3, the sum of the digits of these highlighted numbers will consistently yield a multiple of 3.
This statement is also true because any multiple of 6 is inherently even. Multiples of 6 are formed by multiplying 6 by whole numbers (0, 1, 2, 3, ...), and since 6 is even, all resulting products will also be even.
This statement is true due to the commutative property of multiplication, which states that the order in which numbers are multiplied does not affect the product. This property holds true for all multiplication, including the multiples of 6.
This statement is incorrect because the tens digit does not consistently stay the same for two multiples of 6 before increasing. For example, the multiples of 6 transition through the tens column irregularly, with the tens digit changing at different intervals (e.g., from 6 to 12, the tens digit increases immediately).
This statement is incorrect because not all multiples of 6 are also multiples of 12. For instance, 6 is a multiple of 6 but not of 12, hence not every highlighted number can be guaranteed to be divisible by 12.
The identified correct responses reflect fundamental properties of multiples of 6. They demonstrate the relationships between even numbers and divisibility rules, as well as the commutative property of multiplication. The incorrect options, however, either misrepresent the behavior of the tens digits in the sequence of multiples or make unfounded claims about divisibility by 12. Recognizing these patterns helps students deepen their understanding of multiplication and number properties.
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