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H.C.F. & L.C.M.

1.

Find the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.

Required number = H.C.F. of $$(91 - 43)$$, $$(183 - 91)$$ and $$(183 - 43)$$
= H.C.F. of $$48, 92$$ and $$140 = 4$$.

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2.

The greatest four digit number which is divisible by 15, 25, 40, 75 is

We want 4 digit number, so option A is not the answer

Now we want greatest numnber, So out of remaining options, 9600 is greatest

9600 is divisible by 25, 75, 40, and 15

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3.

Find the H.C.F. of 54, 288, 360

Using factorization method,
$$18 = 2 \times 3^2$$
$$288 = 2^5 \times 3^2$$
$$360 = 2^3 \times 3^2 \times 5$$
So H.C.F. will be minimum term present in all three,
i.e. $$2 \times 3^2 = 18$$

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4.

Find the H.C.F. of
$$2^{2}\times 3^{2}\times 7^{2},2\times 3^{4} \times 7$$

HCF is Highest common factor, so we need to get the common highest factors among given values.
So we get,
$$2 \times 3 \times 3 \times 7 = 126$$

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5.

The L.C.M. of two numbers is 14560 and their H.C.F. is 13. If one of them is 416, the other is

416 X number = 14560 X 13

Therefore, numbr is 455

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6.

A number when divided by 893 the remainder is 193. What will be the remainder when it is divided by 47?

Number is divided by 893. Remainder = 193.

Also, we observe that 893 is exactly divisible by 47,

So now simply divide the remainder by 47

Remainder is 5

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7.

The least number which when divided by the numbers 3, 5, 6, 8, 10 and 12 leaves in each case a remainder 2 but which when divided by 13 leaves no remainder?

No answer description available for this question.

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8.

The greatest number of four digits which is divisible by 15, 25, 40 and 75 is:

Greatest number of 4-digits is 9999.
L.C.M. of 15, 25, 40 and 75 is 600.
On dividing 9999 by 600, the remainder is 399.
Required number,
$$(9999 - 399) = 9600$$

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9.

3 birds fly along the circumference of a jungle. They complete one round in 27 minutes, 45 minutes and 63 minutes respectively. Since they start together, when will they meet again at the starting position?

We need the instance which means the LCM of times of all 3 birds.

Therefore, LCM = 9 X 3 X 5 X 7 = 945 = 945 minutes

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10.

252 can be expressed as a product of primes as:

Clearly, $$252 = 2 \times 2 \times 3 \times 3 \times 7$$