SAT Algebra Student-Produced Response

Algebra Level 3

For all positive integers $k$ , $\Diamond k$ is the sum of all of $k$ 's factors. How many factors does $\Diamond 20 - \Diamond 10$ have?

The answer is 8.

2 solutions

Tatiana Georgieva Staff
Feb 5, 2015

Correct Answer: 8

Solution:

Tip: Follow directions exactly.
Tip: Read the entire question carefully.
The factors of 20 are 1, 2, 4, 5, 10, and 20. Therefore,

$\Diamond 20 = 1 + 2 + 4 + 5 + 10 + 20=42.$

The factors of 10 are 1, 2, 5, and 10. Therefore,

$\Diamond 10 = 1 + 2 + 5 +10=18$

and

$\Diamond 20 - \Diamond 10 = 42-18=24.$

24 has 8 factors: 1, 2, 3, 4, 6, 8, 12, and 24. So, the answer is 8.

Common Mistakes:

Not including 1 and $k$ in the list of $k$ 's factors.
Finding the number of $\Diamond 20'$ s factors, not their sum; and finding the number of $\Diamond 10'$ s factors, not their sum.
Finding the sum of $(\Diamond 20 - \Diamond 10)'$ s factors, not their number.

If you got this problem wrong, you should review SAT Newly Defined Functions and SAT Factors, Divisibility, and Remainders .

Another way to solve this is to notice that $10$ is a factor of $20$ . So we can interpret $\Diamond20 - \Diamond10$ as the sum of factor(s) of $20$ but not a factor of $10$ . By inspection, we can see that $4$ and $20$ are the only numbers that satisfies the criteria. Thus $\Diamond20 - \Diamond10 = 4+20 = 24 = 2^3 \times 3$ . With the number of factors to be $(3+1)\times (1+1) = 8$ .

Pi Han Goh - 5 years, 11 months ago

Anna Anant
Mar 2, 2015

20 - 1,2,4,5,10,20
10 - 1,2,5,10
1+2+4+5+10+20=42
1+2+5+10=18
42-18=24
24 - 1,2,3,4,6,8,12,24
8 factors

SAT Algebra Student-Produced Response

The answer is 8.

2 solutions

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