Problem : 1 / 11
[ HCF_LCM ]
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1 .
The HCF of two numbers is 14 and their LCM is 11592 . If one of the numbers is 504 , find the other?
Solution:
`"Here, HCF"=14, "LCM"=11592, "No1"=504 and "No2"=?` `"No1" * "No2" = "HCF" * "LCM"` `"No2 "= ( "HCF" * "LCM" ) / ("No1") = ( 14 * 11592 ) / 504 = 322`.

Problem : 2 / 11
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2 .
Find the largest number which can exactly divide 513,783,1107 .
Solution:
`"Required number" = "HCF of " 513,783,1107 = 27`. Find HCF of (513,783) 513 | | | | | | | | | | | | | | | |

HCF of (513, 783) = 27 Now find HCF of (27,1107) HCF of (27, 1107) = 27 `:.` HCF of given numbers (513,783,1107) = 27

Problem : 3 / 11
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3 .
Find the smallest number exactly divisible by 12,15,20,27 .
Solution:
`"Required number" = "LCM of " 12,15,20,27 = 540`.2 12 15 20 27 2 6 15 10 27 3 3 15 5 27 3 1 5 5 9 3 1 5 5 3 5 1 5 5 1 1 1 1 1

LCM of 12,15,20,27 = 2 × 2 × 3 × 3 × 3 × 5 = 540

Problem : 4 / 11
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4 .
Find the least number which when divided by 6,7, 8, 9,12 leaves the same remainder 2 in each case.
Solution:
`"Required number" = ( "LCM of " 6,7,8,9,12 ) + 2 = 504 + 2 = 506`.2 6 7 8 9 12 2 3 7 4 9 6 2 3 7 2 9 3 3 3 7 1 9 3 3 1 7 1 3 1 7 1 7 1 1 1 1 1 1 1 1

LCM of 6,7,8,9,12 = 2 × 2 × 2 × 3 × 3 × 7 = 504

Problem : 5 / 11
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5 .
Find the largest number which divides 77 , 147 , 252 to leave the same remainder in each case.
Solution:
`"Required number" = "HCF of " ( 147 - 77 ), ( 252 - 147 ), ( 252 - 77 )` `= "HCF of " 70, 105, 175 = 35`. Factor of 70,105,175 are as follows... 70 = 2 × 5 × 7 105 = 3 × 5 × 7 175 = 5 × 5 × 7 HCF = 5 × 7 = 35 HCF of `70,105,175` is `35`

Problem : 6 / 11
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6 .
Greatest number which can divide 1354 , 1806 , 2762 leaving the same remainder 10 in each case.
Solution:
The number divides 1354 and leaves 10 as remainder `:.` The number exactly divides 1354 - 10 = 1344 The number divides 1806 and leaves 10 as remainder `:.` The number exactly divides 1806 - 10 = 1796 The number divides 2762 and leaves 10 as remainder `:.` The number exactly divides 2762 - 10 = 2752 Now, we have to find HCF of `1344, 1796, 2752` Find HCF of (1344,1796) 1344 | | | | | | | | | | | | | | | | | | | | | | | | | | |

HCF of (1344, 1796) = 4 Now find HCF of (4,2752) HCF of (4, 2752) = 4 `:.` HCF of given numbers (1344,1796,2752) = 4 `:.` Required number = HCF of `1344, 1796, 2752 = 4`.

Problem : 7 / 11
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7 .
The greatest number that will divide 1657 , 2772 leaving respectively 6 , 5 as remainder.
Solution:
The number divides 1657 and leaves 6 as remainder `:.` The number exactly divides 1657 - 6 = 1651 The number divides 2772 and leaves 5 as remainder `:.` The number exactly divides 2772 - 5 = 2767 Now, we have to find HCF of `1651, 2767` Find HCF of (1651,2767) 1651 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |

`:.` HCF of given numbers (1651,2767) = 1 `:.` Required number = HCF of `1651, 2767 = 1`.

Problem : 8 / 11
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8 .
The greatest number that will divide 290 , 460 , 552 leaving respectively 4 , 5 , 6 as remainder.
Solution:
The number divides 290 and leaves 4 as remainder `:.` The number exactly divides 290 - 4 = 286 The number divides 460 and leaves 5 as remainder `:.` The number exactly divides 460 - 5 = 455 The number divides 552 and leaves 6 as remainder `:.` The number exactly divides 552 - 6 = 546 Now, we have to find HCF of `286, 455, 546` Find HCF of (286,455) 286 | | | | | | | | | | | | | | | | | | | |

HCF of (286, 455) = 13 Now find HCF of (13,546) HCF of (13, 546) = 13 `:.` HCF of given numbers (286,455,546) = 13 `:.` Required number = HCF of `286, 455, 546 = 13`.

Problem : 9 / 11
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9 .
The product of HCF and LCM of 18 and 16 is
Solution:
Product of HCF and LCM = product of given numbers `= 18 * 16 = 288`.

Problem : 10 / 11
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10 .
LCM of two numbers is 14 times their HCF. The sum of LCM and HCF is 600 . If one number is 280 , then find the other number ?
Solution:
`"LCM" = 14" HCF"` `"LCM + HCF " = 600` `=> 14 " HCF + HCF " = 600` `=> 15 " HCF " = 600` `=> "HCF " = 600 / 15 = 40` Here, `"LCM" = 14 " HCF"` `:. "LCM " = 14 * 40 = 560`. `:. "Other number " = ( "HCF" * "LCM" ) / ("Given number"} = ( 40 * 560 ) / 280 = 80`.

Problem : 11 / 11
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11 .
Find the least number which is exactly divided by 28,36,45 when we add 19 to it
Solution:
`"Required number" = ( "LCM of " 28,36,45 ) - 19 = 1260 - 19 = 1241`.2 28 36 45 2 14 18 45 3 7 9 45 3 7 3 15 5 7 1 5 7 7 1 1 1 1 1

LCM of 28,36,45 = 2 × 2 × 3 × 3 × 5 × 7 = 1260