Let A = ${n \in N : n i s a 3 - d i g i t n u m b e r}$

B = ${9 k + 2 : k \in N}$

And C = ${9 k + l : k \in N}$ for some $l (0 < l < 9)$

If the sum of all the elements of the set $A \cap (B \cup C)$ is 274 × 400, then $l$ is equal to___________.

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alok kumar singh | Contributor-Level 10

a month ago

Sum of elements $A \cap (B \cup C) = 274 \times 400$

In set B numbers of the form 9k + 2 are {101, 109, .992}

$∴ s u m = \frac{100}{2} (101 + 992) = \frac{100 \times 1093}{2} . . . . . . (i)$

Another possible number is 9k + 5 forms are {104, .995}

$∴ s u m = \frac{100}{2} (104 + 995) = \frac{100}{2} \times 1099 . . . . . . . . . (i i)$

$∴ T o t a l = \frac{100}{2} \times [1093 + 1099] = 100 \times 1096 = 274 \times 4 \times 100 = 274 \times 400$

$∴$ possible value of $l$ = 5

Sum of elements <math> <mrow> <mi>A</mi> <mo>∩</mo> <mrow> <mo>(</mo> <mrow> <mi>B</mi> <mo>∪</mo> <mi>C</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>2</mn> <mn>7</mn> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mn>0</mn> <mn>0</mn> </mrow> </math>  In set B numbers of the form 9k + 2 are {101, 109, .992}  <math> <mrow> <mo>∴</mo> <mi>s</mi> <mi>u</mi> <mi>m</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mn>0</mn> <mn>1</mn> <mo>+</mo> <mn>9</mn> <mn>9</mn> <mn>2</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>9</mn> <mn>3</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math>         Another possible number is 9k + 5 forms are {104, .995}  <math> <mrow> <mo>∴</mo> <mi>s</mi> <mi>u</mi> <mi>m</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mn>0</mn> <mn>4</mn> <mo>+</mo> <mn>9</mn> <mn>9</mn> <mn>5</mn> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>9</mn> <mn>9</mn> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mi>i</mi> </mrow> <mo>)</mo> </mrow> </mrow> </math> <math> <mrow> <mo>∴</mo> <mi>T</mi> <mi>o</mi> <mi>t</mi> <mi>a</mi> <mi>l</mi> <mo>=</mo> <mfrac> <mrow> <mn>1</mn> <mn>0</mn> <mn>0</mn> </mrow> <mrow> <mn>2</mn> </mrow> </mfrac> <mo>×</mo> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mn>0</mn> <mn>9</mn> <mn>3</mn> <mo>+</mo> <mn>1</mn> <mn>0</mn> <mn>9</mn> <mn>9</mn> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>9</mn> <mn>6</mn> <mo>=</mo> <mn>2</mn> <mn>7</mn> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mo>×</mo> <mn>1</mn> <mn>0</mn> <mn>0</mn> <mo>=</mo> <mn>2</mn> <mn>7</mn> <mn>4</mn> <mo>×</mo> <mn>4</mn> <mn>0</mn> <mn>0</mn> </mrow> </math>            <math> <mrow> <mo>∴</mo> </mrow> </math> possible value of <math> <mrow> <mi mathvariant="script">l</mi> </mrow> </math> = 5

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Let A = ${n \in N : n i s a 3 - d i g i t n u m b e r}$

B = ${9 k + 2 : k \in N}$

And C = ${9 k + l : k \in N}$ for some $l (0 < l < 9)$

If the sum of all the elements of the set $A \cap (B \cup C)$ is 274 × 400, then $l$ is equal to___________.

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Let A = { n ∈ N : n i s a 3 − d i g i t n u m b e r } B = { 9 k + 2 : k ∈ N } And C = { 9 k + l : k ∈ N } for some l ( 0 < l < 9 ) If the sum of all the elements of the set A ∩ ( B ∪ C ) is 274 × 400, then l is equal to___________.

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Let A = ${n \in N : n i s a 3 - d i g i t n u m b e r}$

B = ${9 k + 2 : k \in N}$

And C = ${9 k + l : k \in N}$ for some $l (0 < l < 9)$

If the sum of all the elements of the set $A \cap (B \cup C)$ is 274 × 400, then $l$ is equal to___________.