Question

write the missing factor(s). use factors greater than 1.

1. ( 9 \times diamond = 261 )
2. ( diamond \times 14 \times diamond = 294 )
3. ( diamond \times diamond \times 17 = 935 )

list all the factors of each number.

1. 24
2. 88
3. 31
4. circle the prime numbers.

18 19 2 27 49 53
find the greatest common factor.

1. 18, 45
2. 42, 70
3. 24, 60, 84
4. 32, 56, 72

find the lowest common multiple.

1. 3, 7
2. 4, 6
3. 2, 3, 5
4. 4, 7, 9
5. every 4th day the lunchroom students in one school have movie day. every 15th day is pizza day. what is the first day that both events occur on the same day?

Explanation:

Step1: Solve \(9\times\lozenge = 261\)

We know that if \(a\times b = c\), then \(b=\frac{c}{a}\). Here \(a = 9\) and \(c = 261\). So \(\lozenge=\frac{261}{9}=29\)

Step2: Solve \(\lozenge\times14\times\lozenge = 294\)

Let the missing factor be \(x\). Then the equation is \(14x^{2}=294\). First, divide both sides by \(14\): \(x^{2}=\frac{294}{14} = 21\). But we made a mistake. Let's factorize \(294=2\times3\times7\times7\). Since \(14 = 2\times7\), then \(294\div14=21=3\times7\). So the factors are \(3\) and \(7\)

Step3: Solve \(\lozenge\times\lozenge\times17 = 935\)

First, divide \(935\) by \(17\): \(935\div17 = 55\). Then factorize \(55=5\times11\). So the factors are \(5\) and \(11\)

Step4: List factors of \(24\)

We find factors by checking divisibility. \(24\div1 = 24\), \(24\div2=12\), \(24\div3 = 8\), \(24\div4=6\). So the factors are \(1,2,3,4,6,8,12,24\)

Step5: List factors of \(88\)

\(88\div1=88\), \(88\div2 = 44\), \(88\div4=22\), \(88\div8 = 11\). So the factors are \(1,2,4,8,11,22,44,88\)

Step6: List factors of \(31\)

Since \(31\) is a prime number (divisible only by \(1\) and itself), the factors are \(1,31\)

Step7: Circle prime numbers

A prime number has exactly two distinct positive divisors: \(1\) and itself. \(18=2\times9\), \(19\) (prime), \(2\) (prime), \(27 = 3\times9\), \(49=7\times7\), \(53\) (prime). So prime numbers are \(19,2,53\)

Step8: Find GCF of \(18\) and \(45\)

Factorize: \(18=2\times3\times3\), \(45=3\times3\times5\). The common factors are \(3\times3 = 9\)

Step9: Find GCF of \(42\) and \(70\)

Factorize: \(42=2\times3\times7\), \(70=2\times5\times7\). The common factors are \(2\times7=14\)

Step10: Find GCF of \(24\), \(60\) and \(84\)

Factorize: \(24=2^{3}\times3\), \(60=2^{2}\times3\times5\), \(84=2^{2}\times3\times7\). The common factors are \(2^{2}\times3=12\)

Step11: Find GCF of \(32\), \(56\) and \(72\)

Factorize: \(32=2^{5}\), \(56=2^{3}\times7\), \(72=2^{3}\times3^{2}\). The common factors are \(2^{3}=8\)

Step12: Find LCM of \(3\) and \(7\)

Since \(3\) and \(7\) are prime and co - prime, \(LCM(3,7)=3\times7 = 21\)

Step13: Find LCM of \(4\) and \(6\)

Factorize: \(4 = 2^{2}\), \(6=2\times3\). \(LCM(4,6)=2^{2}\times3=12\)

Step14: Find LCM of \(2\), \(3\) and \(5\)

Since \(2\), \(3\) and \(5\) are prime and co - prime, \(LCM(2,3,5)=2\times3\times5 = 30\)

Step15: Find LCM of \(4\), \(7\) and \(9\)

Factorize: \(4=2^{2}\), \(7\) (prime), \(9 = 3^{2}\). \(LCM(4,7,9)=2^{2}\times3^{2}\times7=252\)

Step16: Solve the movie - pizza day problem

We need to find \(LCM(4,15)\). Factorize: \(4 = 2^{2}\), \(15=3\times5\). \(LCM(4,15)=2^{2}\times3\times5=60\)

Answer:

1. \(29\)
2. \(3\), \(7\)
3. \(5\), \(11\)
4. \(1,2,3,4,6,8,12,24\)
5. \(1,2,4,8,11,22,44,88\)
6. \(1,31\)
7. \(19\), \(2\), \(53\)
8. \(9\)
9. \(14\)
10. \(12\)
11. \(8\)
12. \(21\)
13. \(12\)
14. \(30\)
15. \(252\)
16. \(60\)