In this sentence completion question, which auxiliary verb best completes the wish \"_____ you live long!\" to form a natural and correct expression in English?

Introduction / Context:
Sentence completion questions often test your knowledge of standard English expressions, especially those used for wishes, blessings and polite remarks. The phrase you live long is clearly a wish for someone good health and a long life. The task is to decide which auxiliary verb fits naturally at the start of the sentence so that the entire expression becomes a correct and idiomatic way to offer a blessing.

Given Data / Assumptions:

• The incomplete sentence is: _____ you live long!
• The context is a wish or blessing for long life.
• The options are Might, May, Shall and Should.
• The correct answer must create a grammatically correct and idiomatic English expression.

Concept / Approach:
In English, formal wishes and blessings are often expressed using the structure May + subject + verb. Common examples include May you be happy, May God bless you and May all your dreams come true. This use of may expresses a hope or wish. Might can also express possibility but is not normally used to begin a direct blessing. Shall and Should usually express obligation, determination or advice, which does not fit a warm wish. Therefore, May you live long is the correct and natural expression, and may is the right auxiliary verb to choose in this context.

Step-by-Step Solution:
Step 1: Recognise that the sentence is clearly a blessing or wish for someone to live a long life.Step 2: Recall that English uses the structure May + pronoun + verb to express formal wishes.Step 3: Test May you live long in your mind and notice that it sounds natural, polite and idiomatic.Step 4: Consider Might you live long and observe that it sounds unusual and is not used as a blessing in standard English.Step 5: Consider Shall you live long and Should you live long and notice that they suggest obligation or condition rather than a simple wish.Step 6: Conclude that May is the only auxiliary verb that correctly and naturally completes the sentence as a blessing.

Verification / Alternative check:
To double check, you can compare with other standard phrases you have heard or read, such as May you always succeed, May peace be with you and May you have a wonderful day. All of these use may in exactly the same way. Substituting might, shall or should at the beginning of those sentences would make them sound odd or change their meaning. This comparison confirms that May you live long is the correct and idiomatic expression.

Why Other Options Are Wrong:
Might is used to express a weaker possibility or a hypothetical situation and is not typically used to start a blessing aimed at the listener. Shall is usually used for strong intentions or formal rules, as in We shall overcome, and does not function as a wish in this structure. Should often expresses advice or moral duty, for example You should exercise, and does not fit a warm wish such as You live long. Because the sentence clearly represents a blessing, none of these alternatives is appropriate, so options A, C and D are incorrect.

Common Pitfalls:
Some learners confuse may and might in terms of possibility and do not recall that may is the conventional choice for blessings and formal wishes. Others may think of shall as a strong and formal auxiliary and be tempted to use it at the beginning of any serious sounding sentence. To answer such questions reliably, remember that the standard formula for blessings in English exams is May + subject + verb. Once you internalise this pattern, similar questions become very easy.

Final Answer:
The correct sentence is May you live long!, so the right choice is May.