If \$x\$ and \$y\$ are the tens and the units digit, respectively, of the product \$725{,}278×67{,}066\$, what is the value of \$x+y\$ ?\$12\$\$10\$\$8\$\$6\$\$4\$

So, you were trying to be a good test taker and practice for the GRE with PowerPrep online. Buuuut then you had some questions about the quant section—specifically question 9 of Section 4 of Practice Test 1. Those questions testing our knowledge about Integers can be kind of tricky, but never fear, 4476mountvernon.com has got your back!

## Survey the Question

Let’s search the problem for clues as to what it will be testing, as this will help shift our minds to think about what type of math knowledge we’ll use to solve this question. Pay attention to any words that sound math-specific and anything special about what the numbers look like, and mark them on our paper.

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We want to multiply two numbers together and look at what numbers are in the units and tens places, which will involve things we’ve learned about Integers.

## What Do We Know?

Let’s carefully read through the question and make a list of the things that we know.

We will multiply a \$6\$-digit number by a \$5\$-digit numberWe want to know that product’s sum of its units and tens digits

## Develop a Plan

Let’s attack this problem with a top-down approach. We want to know the sum of the units and tens digits of the product of the numbers \$725{,}278·67{,}066\$. There is one solution readily available to us: we could just multiply the numbers together to get the units and tens digits of the product. That does seem like a time-consuming process, especially if our calculator can’t handle such a large product (like, say, the measly GRE calculator provided to us on test day). So let’s try to find another more clever approach to solving this question.

We might remember from when we first learned long multiplication that the units digit of a product is only determined by the units digits of the numbers being multiplied. Also, the tens digit of a product is determined by the units and tens digits of the numbers being multiplied. If we’re curious about why this is, then let’s feel free to look at it in more depth. Otherwise we can just go ahead and solve this question.

## Concept Refresher – Units and Tens Digits for Multiplication

Since we only care about the units and tens digits, perhaps we would be better off looking at numbers specifically based upon their digits, so let’s split one of the numbers up by its digits. We know that each digit represents a factor of \$10\$, so let’s work on splitting up one of the numbers by writing it as the sum of factors of \$10\$. It doesn’t matter which number we do this for, so we’ll arbitrarily select \$725{,}278\$. So let’s first write it as the sum of factors of \$10\$:

\$\$725{,}278 = 8 + 70 + 200 + 5{,}000 + 20{,}000 + 700{,}000\$\$

Now let’s write out the product that we’re looking for using this:

\$\$725{,}278·67{,}066 = 67{,}066·(8 + 70 + 200 + 5{,}000 + 20{,}000 + 700{,}000)\$\$

And we know that we can use distribution to rewrite this as the sum of multiple products, by distributing the \$67{,}066\$ inside the parentheses.

\$\$725{,}278·67{,}066 = 67{,}066·8 + 67{,}066·70 + 67{,}066·200 + …\$\$

We know that when we multiply two numbers together, any \$0\$’s at the end of the numbers will be there in the product. So for example, \$5·700 = 5·7·100 = 3{,}500\$. So for the three products that involve \$200\$, \$5{,}000\$, \$20{,}000\$ and \$700{,}000\$, we know that they will all have \$0\$’s for their units and tens digits. This would also hold true for later when we multiply by \$67{,}066\$, as we could also split this number up by its digits, focusing on the multiplication with the units and tens digits. So they will add nothing to the sum of the units and tens digits for \$725{,}278·67{,}066\$. For these reasons, we’ll split both numbers up keeping the units and tens digits separate from the rest of the number, then multiply them together. Now let’s get back to the question at hand.

## Solve the Question

So we know that we only need to use the tens and units digits in this multiplication to get the tens and units digits of the product, so let’s select the tens and units digits of the two numbers and multiply them together.

\$\$78·66 = 5{,}148\$\$

So the units and tens digits are \$4\$ and \$8\$. Since \$4+8=12\$, the sum of the units and tens digits is \$12\$, and the correct answer is A, \$12\$.

## What Did We Learn

An important lesson to learn from this problem is that we did NOT have to do the entire product just to find the sum of the units and tens digits. So now we know that for future products where we only want to know the units and tens digits, we can just ignore anything beyond the tens digit.

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