Common factors, and the greatest of them

Part a. List the common factors of 28 and 42. Part b. What is the greatest common factor of 28 and 42? Try it first. Pause if you like. A factor pair makes the given number exactly. Start at 1 and look for pairs in order. Start with 28 counters in 1 row of 28. Both numbers in this pair are factors of 28. The same 28 counters make 2 equal rows of 14, with none left over. Next, 4 rows of 7 make 28. Write each factor pair on a separate row. Turn this arrangement: 7 rows of 4 use the same pair again. The pairs have turned round, so stop. Now find the factors of 42. Start its ladder with 1 row of 42. The same counters make 2 equal rows of 21. Both 2 and 21 are factors of 42. Rearrange them into 3 equal rows of 14. Write this pair on the next row. Next, 6 equal rows of 7 make 42. 7 rows of 6 reverse that pair. We have reached the turn again, so both ladders are complete. Read both numbers in each pair. Which numbers appear on both ladders? Circle the shared numbers on both ladders: 1, 2, 7 and 14. These are the common factors. The greatest of these is 14. It divides both 28 and 42 exactly. A factor of just one number is not enough. A common factor must appear on both ladders.

List the common factors of 18 and 30. Try it first. Pause if you like. Build both ladders first. Which factor pairs make 18, and which make 30? Begin 18 with 1 row of 18. Then 2 equal rows of 9 make 18. Next, 3 equal rows of 6 make 18. 6 rows of 3 reverse the same pair. This ladder is complete. Begin the second ladder with 1 row of 30. 2 equal rows of 15 make 30. 3 equal rows of 10 make 30. Next, 5 equal rows of 6 make 30. 6 rows of 5 reverse the pair, so stop. Now compare the two complete ladders. Read both sides of every multiplication sign. Which factors do the ladders share? The numbers on both ladders are 1, 2, 3 and 6. These are the common factors of 18 and 30. A common factor must appear on both ladders.