Question

Find the greatest number which divides \[285\] and \[1249\] leaving remainder $9$and \[7\] respectively.

Answer 1

Hint: We will subtract remainders in the given values respectively. And then we will do prime factorization of both numbers separately. Further we will take the least common number to calculate the HCF of $276\,\,and\,\,1242$.


Complete step by step solution:
The given numbers are $285$and $1249$ and remainders are $9$and $7$respectively. Then new numbers after subtracting the remainders:
$285 - 9 = 276$
$1249 - 7 = 1242$
Then we will find the HCF of $276$ and $1242$ by using prime factorization

$2$	$276$
$2$	$138$
$3$	$69$
$23$	$23$
	$1$
$2$	$1242$
$3$	$621$
$3$	$207$
$3$	$69$
$23$	$23$
	$1$

$276 = 2 \times 2 \times 3 \times 23$
$ = {2^2} \times 3 \times 23$
$1242 = 2 \times 3 \times 3 \times 3 \times 23$
$ = 2 \times {3^3} \times 23$
HCF of $276$and $1242$$ = 2 \times {3^1} \times 23$
$ = 2 \times 3 \times 23$
$ = 6 \times 23$
$ = 138$ $[\because $product of the smallest power of each common prime factor]
Therefore HCF of $285$and $1249$ is $138$.


Note: In these types of questions usually students get puzzled whether to find HCF or LCM. We note that words like larger, highest etc. are keywords mentioned in the question and they give us ideas to find HCF whereas words like smallest, lowest, least etc. give us direction to find the LCM.