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HomeClass 10mathematicsChapter 1Exercise 1.1Q4
| Exercise 1.1

Question 4

Given that HCF(306,657)=9\text{HCF}(306, 657) = 9, find LCM(306,657)\text{LCM}(306, 657).

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SOLUTION

UNDERSTAND THE QUESTION

The question asks to find the Least Common Multiple (LCM) of two numbers, 306 and 657, given their Highest Common Factor (HCF). The core concept required is the fundamental relationship between the HCF and LCM of two numbers, which states that the product of the HCF and LCM of two numbers is equal to the product of the numbers themselves.

PART A

Find LCM(306, 657) given HCF(306, 657) = 9.

STEP 1 — STATE THE RELATIONSHIP BETWEEN HCF AND LCM

For any two positive integers aa and bb, the product of their Highest Common Factor (HCF) and Least Common Multiple (LCM) is equal to the product of the numbers themselves. The formula is:

HCF(a,b)×LCM(a,b)=a×b
\text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b

STEP 2 — IDENTIFY THE GIVEN VALUES

From the problem statement, we have:

a=306b=657HCF(306,657)=9
\begin{aligned}
a &= 306  \\[0.6em]
b &= 657  \\[0.6em]
\text{HCF}(306, 657) &= 9
\end{aligned}

STEP 3 — SUBSTITUTE THE VALUES INTO THE FORMULA

Substitute the known values into the HCF-LCM relationship formula:

9×LCM(306,657)=306×657
9 \times \text{LCM}(306, 657) = 306 \times 657

STEP 4 — CALCULATE THE PRODUCT OF THE TWO NUMBERS

Compute the product 306×657306 \times 657:

306×657=306×(600+50+7)=(306×600)+(306×50)+(306×7)=183600+15300+2142=201042
\begin{aligned}
306 \times 657 &= 306 \times (600 + 50 + 7)  \\[0.6em]
&= (306 \times 600) + (306 \times 50) + (306 \times 7)  \\[0.6em]
&= 183600 + 15300 + 2142  \\[0.6em]
&= 201042
\end{aligned}

STEP 5 — SOLVE FOR LCM

Divide the product by the HCF to isolate the LCM:

LCM(306,657)=306×6579=2010429=22338
\begin{aligned}
\text{LCM}(306, 657) &= \dfrac{306 \times 657}{9}  \\[0.6em]
&= \dfrac{201042}{9}  \\[0.6em]
&= 22338
\end{aligned}

ANSWER

LCM(306,657)=22338
\text{LCM}(306, 657) = 22338

COMMON MISTAKES

Forgetting that the product of HCF and LCM equals the product of the two numbers, and instead trying to find prime factorizations without using the given HCF. Making arithmetic errors when calculating the product 306×657306 \times 657 or when dividing by 9. Confusing the formula and dividing the product of the numbers by the LCM instead of the HCF.

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