10th Maths 1.2

Exercise 1.2

Q1. Express each number as a product of its prime factors:

i) $140$ ii) $156$ iii) $3825$ iv) $5005$ v) $7429$

Sol.

$140=2times2times5times7$

$140=2^2times5^1times7^1$

Answer:

$=2^2times5^1times7^1$ .

ii)

$156=2times2times3times13$

$156=2^2times3times13$

Answer:

$=2^2times3times13$ .

iii)

$3825=3times3times5times5times17$

$3825=3^2times5^2times17$ .

Answer:

$=3^2times5^2times17$ .

iv)

$5005=5times7times11times13$

$5005=5^1times7^1times11^1times13^1$ .

Answer:

$=5^1times7^1times11^1times13^1$ .

$7429=17times19times23$

$7429=17^1times19^1times23$

Answer:

$=17^1times19^1times23$

Q2. Find the $LCM$ and $HCF$ of the following pairs of integers and verify that $LCM×HCF=$ porduct of the two numbers.

i) $26$ and $91$ ii) $510$ and $92$ iii) $336$ and $54$

Sol.

$26=2times13$

$91=7times13$

$HCF=13$

and

$LCM=2times7times13=182$

Verify

$LCMtimesHCF=$ Product of the two numbers

$182times13=26times91$

$2366=2366$

Answer:

$HCF=13$

$LCM=182$

ii)

$510=2times3times5times17$

$92=2times2times23$

$HCF=2$

and

$LCM=2times2times3times5times17times23times=23460$

Verify

$LCMtimesHCF=$ Product of the two numbers

$2times23460=510times92$

$46920=46920$

Answer :

$HCF=2$

$LCM=23460$

iii)

$336=2times2times2times2times3times7$

$54=2times3times3times3$

$HCF=2times3=6$

$LCM=2times2times2times2times3times3times3times7=3024$

Verify

$LCMtimesHCF=$ Product of the two numbers

$3024times6=336times54$

$18144=18144$

Answer :

$HCF=6$

$LCM=3024$

Q3. Find the $LCM$ and $HCF$ of the following integers by applying the prime factorisation method.

i) $12,15$ and $21$ ii) $17,23$ and $29$ iii) $8,9$ and $25$

Sol.

$12=2times2times3$

$15=3times5$

$21=3times7$

$HCF=3$

$LCM=2times2times3times5times7=420$

Answer :

$HCF=3$

$LCM=420$

ii)

$17=1times17$

$23=1times23$

$29=1times29$

$HCF=1$

$LCM=17times23times29=11339$

Answer :

$HCF=1$

$LCM=11339$

iii)

$8=1times2times2times2$

$9=1times3times3$

$25=1times5times5$

$HCF=1$

$LCM=2times2times2times3times3times5times5=1800$

Answer :

$HCF=1$

$LCM=1800$

Q4. Given that $HCF (306, 657)=9$ , find $LCM (306, 657)$ .

Sol. :

$LCMtimesHCF=$ Product of two numbers

$LCMtimes9=306times657$

$LCM=frac(306times657)(9)$

$LCM=frac(201042)(9)$

$LCM=22338$

Answer :

$LCM=22338$

Q5. Check whether $6^n$ can end with the digit $0$ for any natural number $n$ .

Sol. :

We know that

$6^n=2^ntimes3^n$ in which

$5$ is not a prime factor and the number which end with the digit

$0$ is divisible by

$5$ , has

$5$ as prime factor.

Answer :

Thus, the number

$6^n$ cannot end with the digit

$0$ for any natural number.

Q6. Explain why $7times11times13+13$ and $7times6times5times4times3times2times1+5$ are composite numbers.

Sol. :

$=7times11times13+13$

$=13(7times11+1)$

$=13times78$

Which is a composite number

and

$=7times6times5times4times3times2times1+5$

$=5(7times6times4times3times2times1+1)$

$=5(1008+1)$

$=5times1009$

which is a composite number.

Answer :

Thus, the given numbers are composite numbers.

Q7. There is a circular path around a sports field. Sonia takes $18$ minutes to drive one round of the field, while Ravi takes $12$ minutes for the same. Suppose they both start at the same point and at the same time and go in the same direction. After how many minutes will they meet again at the starting point?

Sol.

$18=2times3times3$

$12=2times2times3$

$LCM(18, 12)=2times2times3times3times=36$

Answer :

Thus, they will meet again at the starting point after

$36$ minutes.

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