HCF and LCM Problems

16, 24
Factor of 16 = 2 × 2 × 2 × 2
Factor of 24 = 2 × 2 × 2 × 3
HCF = 8

Factors of 375 = 5 × 5 × 5 × 3
Factors of 625 = 5 × 5 × 5 × 5
Factor of 875 = 5 × 5 × 5 × 7
HCF = 5 × 5 × 5 = 125

This is a GCD problem. To find the widest possible strips that both 70-inch and 90-inch pieces can be cut into, find GCD(70, 90). Prime factors: 70 = 2 × 5 × 7, 90 = 2 × 3² × 5. Common factors: 2 × 5 = 10 inches.

54, 78
Factors of 54 = 2 × 3 × 3 × 3
Factors of 78 = 2 × 3 × 13
LCM = 2 × 3 × 3 × 3 × 13 = 702

12, 15, 24
Factors of 12 = 2 × 2 × 3
Factors of 24 = 2 × 2 × 2 × 3
Factors of 15 = 3 x 5
HCF = 3

This is an LCM problem. A caller gets both prizes when they are both a 100th caller and a 40th caller. Find LCM(100, 40). Prime factors: 100 = 2² × 5², 40 = 2³ × 5. LCM = 2³ × 5² = 8 × 25 = 200.

Both riders start together and move in the same direction. They'll meet again at the starting point when both have completed whole number of laps. Find LCM(12, 16). Prime factors: 12 = 2² × 3, 16 = 2⁴. LCM = 2⁴ × 3 = 48 minutes.

7, 9
LCM of 7 and 9 = 7 × 9 = 63

Sally has 5-inch pieces and Chloe has 7-inch pieces. To build equal length tracks, find the smallest length divisible by both 5 and 7. Since 5 and 7 are co-prime, LCM(5, 7) = 35 inches. Sally needs 7 pieces (35 ÷ 5) and Chloe needs 5 pieces (35 ÷ 7).

David exercises every 11th day (cycle of 11) and Patrick every 5th day (cycle of 5). They exercise together today and will again when both cycles complete. Find LCM(11, 5) = 55. They'll exercise together again on the 55th day.

Multiples of 7 = 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84
Multiples of 12 = 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
Common multiples of 7 and 12 = 84
Lowest common multiple of 7 and 12 is 84.

Jamie has 55 apple trees and 22 peach trees. To have same number of each tree type per row with maximum trees, find GCD(55, 22). 55 = 5 × 11, 22 = 2 × 11. GCD = 11 trees per row (5 rows of apple trees, 2 rows of peach trees).

Factors of 74 = 1, 2, 37, 74
Factors of 84 = 1, 84, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42
Common factor = 1, 2
Highest common factor of 74 and 84 is 2.

Multiples of 8 = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80
Multiples of 14 = 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154
Lowest common multiple is 56.

Multiple of 14 = 14, 28, 42, 56, 70, 84, 98, 112, 126, 140
Multiples of 18 = 18, 36, 54, 90, 108, 126, 144, 162, 180
Lowest common multiple of 14 and 18 = 126

Factors of 240 = 1, 2, 3, 4, 6, 8, 10, 12, 15, 16, 20, 30, 24, 40, 60, 80, 120, 240, 5, 48
Factors of 196 = 1, 2, 4, 7, 14, 28, 49, 88, 196
Common factors = 1, 2, 4
HCF of 240 and 196 = 4

Factors of 120 = 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60
Factors of 160 = 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 80, 160
Common factor = 1, 2, 3, 5, 8, 10, 20, 40
HCFof 120 and 160 is 40.

Prime Factors of 19 = 1 x 19
Prime Factors of 21 = 3 x 7
LCM of 19 and 21 = 19 x 3 x 7 = 399

Factors of 156 = 1, 2, 3, 4, 6, 12, 13, 36, 39, 53, 78, 156
Factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Common factors = 1, 2, 3, 4, 6, 12
Highest common factors of 156 and 96 is 12.

Factors of 144 = 1, 2, 3, 4, 6, 9, 12, 16, 18, 24, 36, 48, 72, 144
Factors of 164 = 1, 2, 4, 41, 82, 164
Common factors = 1, 2, 4
HCF = 4

Prime Factors of 16 = 2 x 2 x 2 x 2
Prime Factors of 22 = 2 x 11
Lowest common multiple = 2⁴ x 11 = 176

Prime Factors of 23 = 23
Prime Factors of 26 = 2 x 13
LCM of 23 and 26 = 2 x 13 x 23 = 598

Factors of 96 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96
Factors of 118 = 1, 2, 59, 118
Common factors = 1 and 2
HCF of 96 and 118 = 2

Factors of 230 = 1, 2, 5, 10, 23, 46, 115, 230
Factors of 148 = 1, 2, 4, 37, 74, 148
Common factors of 230 and 148 are 1, 2.
HCF of 230 and 148 is 2.

Multiples of 6 are 6, 12, 18, 24, 36, 42, 48, 54, 60....
Multiples of 9 are 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.......
Common multiples = 18, 36, 54
LCM of 6 and 9 is 18.