6.NS Adding Multiples
Nina was finding multiples of 6. She said,
18 and 42 are both multiples of 6, and when I add them, I also get a multiple of 6: $$18+42 = 60.$$Explain to Nina why adding two multiples of 6 will always result in another multiple of 6.
Solutions
18 and 42 are both multiples of 6 which means 6 is a factor of both numbers. $$6 \cdot 3 = 18$$ $$ 6 \cdot 7 = 42$$
When you add 18 and 42, you can use the distributive property to show that the sum is also a multiple of 6: $$6 \cdot 3 + 6 \cdot 7 = 6 \cdot (3 + 7) = 6 \cdot 10$$
If you have any two multiples of 6, you can write them both as 6 times a whole number. Using the distributive property, their sum will also be 6 times a whole number, just like in the example of 18 and 42.
18 and 42 are both multiples of 6 which means 6 is a factor of both numbers. $$6 \cdot 3 = 18$$ $$ 6 \cdot 7 = 42$$ When you add 18 and 42, you can use the distributive property to show that the sum is also a multiple of 6: $$6 \cdot 3 + 6 \cdot 7 = 6 \cdot (3 + 7) = 6 \cdot 10$$ If you have any two multiples of 6, you can write them both as 6 times a whole number. We can write this in symbols as $$6\cdot a$$ $$6\cdot b$$ (where \(a\) and \(b\) are both whole numbers). When you add them, you have $$ 6\cdot a + 6\cdot b = 6 \cdot (a+b)$$ Since \(a\) and \(b\) are whole numbers, so is \(a+b\). So the sum of two multiples of 6 is always a multiple of 6.
Please note that there is nothing special about 6 in this argument, so it will work exactly the same way with any whole number in place of 6.