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9. Find the smallest number which when increased by 20 is exactly divisible by 90 and 144.

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To find the smallest number that is divisible by both 90 and 144 when increased by 20, we need to find the least common multiple (LCM) of 90 and 144. Then, we'll add 20 to that LCM to get our answer. First, let's find the LCM of 90 and 144. The prime factorization of 90 is 2×32×52×32×5. The...
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To find the smallest number that is divisible by both 90 and 144 when increased by 20, we need to find the least common multiple (LCM) of 90 and 144. Then, we'll add 20 to that LCM to get our answer.

First, let's find the LCM of 90 and 144.

The prime factorization of 90 is 2×32×52×32×5.

The prime factorization of 144 is 24×3224×32.

To find the LCM, we take the highest power of each prime factor that appears in either number:

LCM=24×32×5=720LCM=24×32×5=720

Now, we add 20 to 720:

720+20=740720+20=740

So, the smallest number that, when increased by 20, is exactly divisible by both 90 and 144 is 740.

 
 
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Find the HCF of 52 and 117 and express it in form 52x + 117y.
From Euclid's division lemma,we know that a=bq+r,Since 117>52 ,we can take a=117 & b=52 Now,117=52*2+13(52 is divisor) so,52=13*4+0,the division process stops here,as the remainder becomes 0. Hence...
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Euclid's Division Lemma
Euclid's Division Lemma: Given positive integers a and b,there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b.Euclid’s division algorithm is based on this Lemma. Example:...

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