Dividing by any common factor

The idea

In the common division procedure the numbers need not be divided by one prime at a time; they can be divided by any common factor that can be identified.

A shortcut?

Guna wants to find the HCF of 300 and 150. She says: “Both are multiples of 50. I will divide both by 50 straight away.”

Is 50 allowed in the ladder, even though 50 is not a prime?

• Yes, it divides both numbers
• No, only primes are allowed
• Only if it ends in 0

Big steps, same answer

Divide both numbers by any common factor from the buttons, including big ones. Finish the ladder, read the HCF and LCM, then try a different route.

Any common factor will do

A common factor such as 50 is a bundle of primes (50 = 2 × 5 × 5). Dividing by it does several prime steps at once.

HCF = 50 × 3 = 150. LCM = 50 × 3 × 2 × 1 = 300.

Anshu: 630 and 770 ÷ 10 = 63 and 77; ÷ 7 = 9 and 11. HCF = 10 × 7 = 70. LCM = 10 × 7 × 9 × 11 = 6930.

In the common division procedure, the numbers need not be divided by one prime at a time. They can be divided by any common factor that you can see. The HCF is the product of the divisors; the LCM is that times the final quotients.

• Guna: 300 and 150
• 300, 150
• 50 | 6, 3
• 3 | 2, 1

Notes

The numbers can be divided by any common factor that you can identify, not only one prime at a time. HCF = the product of the divisors. LCM = that product times the final quotients.

Check yourself

Use one big step. 90 and 150 are both divided by 30, giving 3 and 5. What is the HCF?

Answer: 30

One divisor, 30. The HCF is 30. The LCM is 30 × 3 × 5 = 450.

84 and 132 are both divided by 12, giving 7 and 11. What is the LCM?

Answer: 924

HCF = 12. LCM = 12 × 7 × 11 = 924.

Anshu divides 630 and 770 by 10 and gets 63 and 77. Is the ladder finished?

A ladder for 48 and 72 divides by 12 and then by 2, giving 2 and 3. Is the HCF 12?

• Yes, 63 and 77 are not equal.. The test is not whether they are equal. It is whether they share a prime.
• Yes, we stop after one division.. We must keep going while the numbers still share a factor. 63 and 77 share 7.
• No, both are multiples of 7 (63 = 7 × 9, 77 = 7 × 11). Divide again by 7. — correct. Yes! Only when the two numbers share no prime do we stop. After 7 we reach 9 and 11.
• Yes, the first divisor is always the HCF.. Not unless the ladder ends there. After 12 we still had 4 and 6, which share 2.
• No. The HCF is 12 × 2 = 24: we multiply all the divisors. — correct. Yes! 12 × 2 = 24. And the LCM is 24 × 2 × 3 = 144.
• No. The HCF is 2 × 3 = 6.. 2 and 3 are the final quotients. The HCF is the product of the divisors.