Question:
If the number \(7x342y\) is divisible by \(88\), then \((x,y)=\ ?\)
If the number \(7x342y\) is divisible by \(88\), then \((x,y)=\ ?\)
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For divisibility by \(88\), always check divisibility by \(8\) and \(11\) separately.
Updated On: Jun 12, 2026
- \((1,4)\)
- \((4,4)\)
- \((4,6)\)
- \((6,8)\)
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The Correct Option is B
Solution and Explanation
Concept:
Since
\[
88=8\times11,
\]
the number must be divisible by both \(8\) and \(11\).
Step 1: Apply divisibility by 8. The last three digits are \[ 42y \] Testing the options: \[ 424\div8=53 \] Hence \(y=4\) works. \[ 426 \] is not divisible by \(8\). \[ 428 \] is not divisible by \(8\). Therefore, \[ y=4 \]
Step 2: Apply divisibility by 11. The number becomes \[ 7x3424 \] For divisibility by 11, \[ (7+3+2)-(x+4+4) \] must be a multiple of \(11\). \[ 12-(x+8) \] \[ =4-x \] This must be \(0\) or \(\pm11\). Thus, \[ 4-x=0 \] \[ x=4 \] Hence, \[ (x,y)=(4,4) \] Therefore, \[ \boxed{(4,4)} \]
Step 1: Apply divisibility by 8. The last three digits are \[ 42y \] Testing the options: \[ 424\div8=53 \] Hence \(y=4\) works. \[ 426 \] is not divisible by \(8\). \[ 428 \] is not divisible by \(8\). Therefore, \[ y=4 \]
Step 2: Apply divisibility by 11. The number becomes \[ 7x3424 \] For divisibility by 11, \[ (7+3+2)-(x+4+4) \] must be a multiple of \(11\). \[ 12-(x+8) \] \[ =4-x \] This must be \(0\) or \(\pm11\). Thus, \[ 4-x=0 \] \[ x=4 \] Hence, \[ (x,y)=(4,4) \] Therefore, \[ \boxed{(4,4)} \]
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