1. What is the largest integer that is a factor of all three of the following numbers: 2160, 1344, 1440?
a) 6 b) 8 c) 12 d)16 e) none of these
2. The annual income of a family is budgeted as follows:
1/10 for clothing, 1/3 for food, and 1/5 for rent.
This leaves $1320 for other expenses and savings. Find the annual income.
a) $2156 b) $3600 c) $23,760
d) $39,600 e) none of these answers
To find the largest integer that is a factor of all three numbers 2160, 1344, and 1440, we can determine the factors of each number and find their greatest common factor (GCF).
Step 1: Find the factors of each number:
- Factors of 2160: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 80, 90, 108, 120, 135, 180, 216, 270, 360, 540, 720, 1080, 2160
- Factors of 1344: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 448, 672, 1344
- Factors of 1440: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 288, 360, 480, 720, 1440
Step 2: Find the common factors among all three numbers:
- Common factors of 2160, 1344, and 1440: 1, 2, 3, 4, 6, 8, 12, 24
Step 3: Determine the largest common factor: 24
Therefore, the largest integer that is a factor of all three numbers is 24. Option e) none of these answers.
To find the annual income in question 2, we can set up an equation.
Step 1: Assign variables:
Let x represent the annual income.
Step 2: Set up the equation based on the given information:
x - (1/10)x - (1/3)x - (1/5)x = $1320
Step 3: Simplify the equation:
(10/10)x - (1/10)x - (3/10)x - (2/10)x = $1320
(4/10)x = $1320
(2/5)x = $1320
Step 4: Solve for x:
x = ($1320) / (2/5)
x = ($1320) * (5/2)
x = $3300
Therefore, the annual income is $3300. Option e) none of these answers.
1. To find the largest integer that is a factor of all three numbers (2160, 1344, 1440), we can use the concept of finding the greatest common divisor (GCD) or highest common factor (HCF).
One approach is to list the prime factorization of each of the three numbers and look for the common factors. Then, we can determine the largest integer that appears in all three factorizations.
Prime factorizations:
2160 = 2^4 * 3^3 * 5
1344 = 2^5 * 3 * 7
1440 = 2^5 * 3^2 * 5
Looking at the prime factorizations, we can identify the common factors as follows:
- The common factor of 2 is 2^4 = 16
- The common factor of 3 is 3^1 = 3
- The common factor of 5 is 5^1 = 5
To find the largest integer that appears in all three factorizations, we multiply these common factors: 16 * 3 * 5 = 240.
Therefore, the largest integer that is a factor of all three numbers is 240. Since none of the provided options match this answer, the correct choice would be e) none of these.
2. To find the annual income, we need to solve the equation using the given information.
Let's assign the annual income as "x."
According to the budget:
1/10 of x is allocated for clothing, which can be expressed as x/10.
1/3 of x is allocated for food, which can be expressed as x/3.
1/5 of x is allocated for rent, which can be expressed as x/5.
Adding these three allocations together, we get:
x/10 + x/3 + x/5 = (1320 + x)
To solve this equation, we can find a common denominator. In this case, it is 30. Multiplying each term by 30, we have:
3x + 10x + 6x = 39600 + 30x
19x = 39600 + 30x
19x - 30x = 39600
-11x = 39600
x = -39600/-11
x = 3600
Therefore, the annual income is $3600. Hence, the correct choice is b) $3600.
2160 = 2*2*2*2*3*3*3*5
1344 = 2*2*2*2*2*2*3*7
1440 = 2*2*2*2*2*3*3*5
HCF = 2*2*2*2*3
= 24
2.
let their income be x
x/10 + x/3 + x/5 + 1320 = x
times 150 , the LCM
15x + 50x + 30x + 198000 = 150x
198000 = 55x
x = 3600
1344 = 1*1344 = 1*2*672 = 1*2*3*224 =
1*2*3*4*56 = 1*2*3*4*7*8.
1440 = 1*1440 = 1*2*720 = 1*2*3*240 =
1*2*3*4*60 = 1*2*3*4*5*12=1*2*3*4*5*6*2
2160 = 1*2160 = 1*2*1080 = 1*2*3*360 =
1*2*3*4*90 = 1*2*3*4*5*18=1*2*3*4*5*6*3
GCF = 4.
2. Income = $X.
(1/10+1/3+1/5)X + 1320 = X
(3/30+10/30*6/30)X + 1320 = x
19x/30 + 1320 = x
Multiply both sides by 30:
19x + 39,600 = 30x
-11x = -39,600
X = $3600