Every day Sally saved a penny, a dime, and a quarter. What is the least number of days required for her to save an amount equivalent to an integral number of dollars?
.36 dollars a day
100 days = 36 dollars
50 days = 18 dollars
25 days = 9 dollars
humm, no more common factors
25. The sum of a penny, a dime, and
a quarter is 36 cents. The number 36
factors into 2²•3². To get an integral
value in dollars, we must multiply by 5² so that the product ends in two zeroes.
Any multiple of 25 will work, but 25 is
the least of those values.
To find the least number of days required for Sally to save an amount equivalent to an integral number of dollars, we can start by figuring out how much money she saves each day.
Let's look at the three coins:
- 1 penny is equal to $0.01
- 1 dime is equal to $0.10
- 1 quarter is equal to $0.25
Now, since Sally saves one penny, one dime, and one quarter each day, her total savings per day would be $0.01 + $0.10 + $0.25 = $0.36.
We want to find the least number of days required to save an amount equivalent to an integral number of dollars. This means we need to find the least number of days required to save $1.00 or 100 cents.
To do this, we need to find the least common multiple (LCM) of 36 (the savings per day) and 100 (the target savings). The LCM will give us the smallest number of days required to reach the target.
The LCM of 36 and 100 can be found by dividing their product by their greatest common divisor (GCD). So, we need to find the GCD first.
The GCD of 36 and 100 can be found using various methods like Euclidean algorithm or prime factorization. Let's use the simpler method of prime factorization here:
- Prime factorization of 36: 2^2 * 3^2
- Prime factorization of 100: 2^2 * 5^2
Now, we take the highest power of each prime factor that appears in either factorization:
- Highest power of 2: 2^2
- Highest power of 3: 3^2
- Highest power of 5: 5^2
Multiplying these highest powers, we get: 2^2 * 3^2 * 5^2 = 2^2 * 9 * 25 = 900
So, the GCD of 36 and 100 is 900.
Now, we can find the LCM by dividing the product of the two numbers by their GCD:
LCM = (36 * 100) / 900 = 3600 / 900 = 4
Therefore, the least number of days required for Sally to save an amount equivalent to an integral number of dollars is 4.