Some Useful Short-Cut Methods

1. H.C.F. and L.C.M. of Decimals
Step 1 Make the same number of decimal places in all the given numbers by suffixing zero(s) if necessary.
Step 2 Find the H.C.F./L.C.M. of these numbers without decimal.
Step 3 Put the decimal point (in the H.C.F./L.C.M. of step 2) leaving as many digits on its right as there are in each
of the numbers

2. L.C.M. and H.C.F. of Fractions

L.C.M = L.C.M. of the numbers in numerators/H.C.F. of the numbers in denominators

H.C.F. = H.C.F. of the numbers in numerators/L.C.M. of the numbers in denominators

3. Product of two numbers
= L.C.M. of the numbers  x H.C.F. of the numbers

4. To find the greatest number that will exactly divide x, y and z.
Required number = H.C.F. of x, y and z.

5. To find the greatest number that will divide x, y and z leaving remainders a, b and c, respectively.
Required number = H.C.F. of (x – a), (y – b) and (z – c).

6. To find the least number which is exactly divisible by x, y and z.
Required number = L.C.M. of x, y and z.

7. To find the least number which when divided by x, y and z leaves the remainders a, b and c, respectively. It is always observed that (x – a) = (y – b) = (z – c) = k (say)
\ Required number = (L.C.M. of x, y and z) – k.

8. To find the least number which when divided by x, y and z leaves the same remainder r in each case.
Required number = (L.C.M. of x, y and z) + r.

9. To find the greatest number that will divide x, y and z leaving the same remainder in each case.
(a) When the value of remainder r is given:
Required number = H.C.F. of (x – r), (y – r) and (z – r).
(b) When the value of remainder is not given:
Required number = H.C.F. of ç(x – y)ç, ç(y – z)ç and ç(z – x)ç

10. To find the n-digit greatest number which, when divided by x, y and z.
(a) leaves no remainder (i.e. exactly divisible)

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